Classification of Algebraic Varieties

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Author: Gerard van der Geer | Carel Faber | Eduard Looijenga | Editors

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EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowro´nski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.)

Classification of Algebraic Varieties Carel Faber Gerard van der Geer Eduard Looijenga Editors

Editors: Carel Faber Department of Mathematics Royal Institute of Technology SE-100 44 Stockholm Sweden

Gerard van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Postbus 94248 1090 GE Amsterdam The Netherlands

Eduard J.N. Looijenga Mathematisch Instituut Universiteit Utrecht Postbus 80010 3508 TA Utrecht The Netherlands

2010 Mathematics Subject Classification: 14E, 14E30, 14D, 14J10 Key words: Classification of algebraic varieties, minimal model program, birational geometry

ISBN 978-3-03719-007-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Printing and binding: Druckhaus Thomas Müntzer GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Stable varieties with a twist D. Abramovich and B. Hassett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Basic properties of log canonical centers F. Ambro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces I. Bauer and F. Catanese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Minimal models, ﬂips and ﬁnite generation: a tribute to V.V. Shokurov and Y.-T. Siu C. Birkar and M. P˘ aun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Remarks on an Example of K. Ueno F. Campana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Special Orbifolds and Birational Classiﬁcation: a Survey F. Campana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Birational geometry of threefolds J.A. Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Emptiness of homogeneous linear systems with ten general base points C. Ciliberto, O. Dumitrescu, R. Miranda, and J. Ro´e . . . . . . . . . . . . . . . . . . . . . . . 189 Finite generation of adjoint rings after Lazi´c: an introduction A. Corti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Log canonical thresholds on varieties with bounded singularities T. de Fernex, L. Ein, and M. Mustat¸a ˘ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Brill-Noether geometry on moduli spaces of spin curves G. Farkas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 On the Bimeromorphic Geometry of Compact Complex Contact Threefolds K. Frantzen and T. Peternell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

vi

Contents

Introduction to the theory of quasi-log varieties O. Fujino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 On Kawamata’s theorem O. Fujino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Remarks on the cone of divisors Y. Kawamata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .317 p-elementary subgroups of the Cremona group of rank 3 Y. Prokhorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Introduction

The quest for understanding the structure of algebraic varieties is a very natural one. In the 19th century it was realized that the birational classiﬁcation of algebraic curves consists in two steps: an irreducible curve has a unique smooth model and its genus g is a discrete invariant, and for ﬁxed genus g the isomorphism classes of curves depend (for g > 1) on 3g − 3 parameters, the moduli. Classiﬁcation of varieties goes back to the attempt of the Italian school around the turn of the 20th century to extend this to dimension 2 and to understand algebraic surfaces. Castelnuovo indicated how one could obtain a minimal model of a surface and showed that the model is unique if the surface is not ruled. Armed with this, Enriques set up a rough classiﬁcation of surfaces that was based on the behaviour of the pluri-canonical systems. Deservedly it is viewed as the main achievement of the Italian School of algebraic geometry around the beginning of the 20th century. In the 1960s Kodaira re-examined this classiﬁcation and put it in a modern framework using the newly obtained techniques of algebraic and complex analytic geometry. He extended the classiﬁcation to compact complex analytic surfaces. The work of Kodaira was continued by Kawai, Iitaka, Ueno and others. A landmark from that time was Hironaka’s 1964 theorem on the existence of a smooth model for any complex algebraic variety. The case of surfaces suggested to look for nefness of the canonical class as a guiding principle. But it was soon realized that the theory in higher dimensions is much more intricate than in dimension 2. The reason was the existence of smooth varieties not possessing a smooth model with a nef canonical class. Thus one had to deal with singularities. It slowly emerged in the 1970s and the early 1980s in work of Reid, Kawamata, Mori and others that by restricting the type of singularities that minimal models were allowed to have one could still build a useful theory of such models. The Minimal Model Program is the quest for a minimal or simplest projective variety that is birational to a given variety and is based on the expectation that there exists a model with limited singularities and nef canonical class or a morphism to a lower-dimensional variety with ample relative anti-canonical class. Contrary to dimension 2, contractions alone should not suﬃce to reach a minimal model, but other birational transformations, the ﬂips, are required. The work of Mori in the 1980s analyzed threefolds for which the canonical bundle is not nef. An important role was played by Mori’s cone of curves, based on his famous Bend and Break theorem that he had developed in 1979. Shokurov showed the termination of ﬂips in the three-dimensional case and in 1988 Mori completed this by his theorem on the existence of ﬂips for three-dimensional varieties, using earlier work of Kawamata and Tsunoda. In this way the combined eﬀorts of Reid, Mori, Kawamata, Koll´ ar, Shokurov and others led to a generalization of the work of the Italian school to dimension 3.

viii

Introduction

The existence of ﬂips and the termination of the process formed the main obstacles for progress in arbitrary dimension. The existence of ﬂips is closely linked to the ﬁnite generation of the canonical ring. From 2005 onwards, great progress has been achieved, ﬁrst by Hacon and McKernan, whose work was then complemented by Birkar and Cascini. Besides the existence of ﬂips in arbitrary dimension using techniques of Shokurov and Siu, this has led to the cornerstone result that for any smooth projective variety the canonical ring is ﬁnitely generated. It was this wave of breakthroughs and surprising new developments that motivated us to devote a conference to the theme “Classiﬁcation of Varieties.” The conference can be seen as a successor of the Texel conferences, though it did not take place on Texel Island, but on another island, Schiermonnikoog. It was held in the second week of May 2009. The present volume is published on the occasion of the conference, although most contributions are not related to lectures given at the conference. It gives ample evidence of the progress that is now being made. We would like to take the opportunity to thank the participants and the speakers, who made the conference a success. We thank the institutions that ﬁnanced the conference: the Foundation Compositio Mathematica and NWO. July 2010

Carel Faber Gerard van der Geer Eduard Looijenga

Stable varieties with a twist Dan Abramovich and Brendan Hassett∗

Abstract. We describe a new approach to the deﬁnition of the moduli functor of stable varieties. While there is wide agreement as to what classes of varieties should appear, the notion of a family of stable surfaces is quite subtle, as key numerical invariants may fail to be constant in ﬂat families. Our approach is to add natural stack structure to stable varieties. For example, given a canonical model we take the global-quotient stack structure arising from its realization as Proj of the canonical ring. Deformations of the stack structure preserve key numerical invariants of the stable variety, including the top self-intersection of the canonical divisor. This approach yields a transparent construction of the moduli stack of stable varieties as a global quotient of a suitable Hilbert scheme of weighted projective stacks. 2010 Mathematics Subject Classiﬁcation. Primary 14J10; Secondary 14D22, 14E30. Keywords. Moduli spaces, Gorenstein singularities, varieties of general type, quotient stacks.

Contents 1 Introduction

1

2 Cyclotomic stacks and weighted projective stacks

6

3 Moduli of stacks with polarizing line bundles

15

4 Moduli of polarized orbispaces

20

5 Koll´ ar families and stacks

24

6 Semilog canonical singularities and compactiﬁcations

29

A The semilog canonical locus is open

33

1. Introduction 1.1. Moduli of stable varieties: the case of surfaces. In the paper [KSB88], Koll´ ar and Shepherd-Barron introduced stable surfaces as a generalization of sta∗ Research of D.A. partially supported by NSF grants DMS-0335501 and DMS-0603284. Research of B.H. partially supported by NSF grants DMS-0196187, DMS-0134259, and DMS0554491 and by the Sloan Foundation.

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D. Abramovich and B. Hassett

ble curves. This class is natural from the point of view of the minimal model program, which shows that any one-parameter family of surfaces of general type admits a unique stable limit. Indeed, the stable reduction process of Deligne and Mumford can be interpreted using the language of minimal models of surfaces. Stable surfaces admit semilog canonical singularities. Codimension-one semilog canonical singularities are nodes, but in codimension two more complicated singularities arise. However, these singularities are all reduced, satisfy Serre’s condition S2 (hence are Cohen–Macaulay in codimension 2), and admit a well-deﬁned Q-Cartier canonical divisor KS . Stable varieties in any dimension are similarly deﬁned as proper varieties with semilog canonical singularities and ample canonical divisor. Koll´ ar and Shepherd-Barron proposed the moduli space of stable surfaces as the natural compactiﬁcation of the moduli space of surfaces of general type. Much of this was established in [KSB88], which also oﬀered a detailed classiﬁcation of the singularities that arise. Boundedness of the class of stable surfaces with ﬁxed invariant (KS )2 was shown by Alexeev [Ale94]. 1.2. The problem of families. To a great extent this established the moduli space of stable surfaces as the ‘right’ geometric compactiﬁcation of moduli of surfaces of general type. But one nagging question remains unresolved: what ﬂat families π : X → B of stable surfaces should be admitted in the compactiﬁcation? The problem stems precisely from the fact that the dualizing sheaf ωS = OS (KS ) is not necessarily invertible for a stable surface. Rather, for a suitable positive integer n depending on S, the sheaf [n]

OS (nKS ) = ωS

:= (ωSn )∗∗

is invertible. As it turns out, there exist ﬂat families π : X → B over a smooth [n] curve where the ﬁbers are stable surfaces but ωπ is not locally free for any n K= 0. For a local example, see Pinkham’s analysis of the deformation space of the cone over a rational normal quartic curve described in [KSB88, Example 2.8]. Two approaches were suggested to resolve this issue (see [HK04] for detailed discussion). In [Kol90, 5.2], Koll´ ar proposed what has come to be called a Koll´ ar family of stable surfaces, where the formation of every saturated power of ωπ is required to commute with base change. Viehweg [Vie95] suggested using families where some saturated power of ωπ is invertible (and in particular, commutes with base extension), i.e., there exists an integer n > 0 and an invertible sheaf on X [n] restricting to ωS on each ﬁber S of π. It has been shown in [HK04] that Viehweg families give rise to a good moduli space. The purpose of this article is to address the case of Koll´ar families. While this question has been considered before (cf. [Hac04]), the solution we propose is perhaps more natural than previous approaches, and has implications beyond the question at hand. We must note that Koll´ar’s [Kol08] resolves the issue by tackling it head on using the moduli spaces of Husks. Our approach goes by way of showing that, in the situation at hand, if we consider objects with an appropriate stack structure,

3

Stable varieties

the problem does not arise. This does require us to show that moduli spaces of such stack structures are well behaved. 1.3. Canonical models and quotient stacks. Here is the basic idea: Let X be a smooth projective variety of general type. A fundamental invariant of X is the canonical ring n R(X) := ⊕n≥0 Γ(X, ωX ), a graded ring built up from the diﬀerential forms on X. This ring has long been known to be a birational invariant of X and has recently been shown to be ﬁnitely generated [BCHM10]. This was known classically for curves and surfaces, and established for threefolds by S. Mori in the 1980’s. The associated projective variety X can := Proj R(X) = (Spec R(X) 1 0)/Gm [n]

is called the canonical model of X. Some saturated power ωX can is ample and invertible, essentially by construction. Therefore, deformations of canonical models (and stable surfaces) are most naturally expressed via deformations of the canonical ring. In these terms, our main problem is that certain deformations of a stable surface do not yield deformations of its canonical ring. Our solution is to replace X can with a more sophisticated geometric object that remembers its algebraic origins. Precisely, we consider quotient stacks X can = ProjR(X) := [(Spec R(X) 1 0)/Gm ] , which admit nontrivial stack-structure at precisely the points where ωX can fails to be invertible. Our main goal is to show that this formulation is equivalent to Koll´ ar’s, and yields a workable moduli space. 1.4. Moduli of stacks: the past. The idea that families with ﬁbers having stack structure should admit workable moduli spaces is not new. On the level of deformations, Hacking’s approach in [Hac04] uses precisely this idea. The case of stable ﬁbered surfaces, namely stable surfaces ﬁbered over curves with semistable ﬁbers, was discussed in [AV00]. It was generalized to many moduli problems involving curves in [AV02] and to varieties of higher dimension with suitable “pluriﬁbration” structures [Abr02]. The ﬁbered and pluriﬁbered examples are a little misleading, since the existence of canonical models and complete moduli in those cases are an outcome of [AV02] and do not require the minimal model program. Still they do reinforce the point that stacks do not pose a true obstacle in deﬁning moduli. We show in this paper that this is indeed the case for stacks of the form X = ProjR for R a ﬁnitely generated graded ring, and moreover this works in exactly the same way it works for varieties, if one takes the appropriate viewpoint. Classical theory of moduli of projective varieties starts by considering varieties embedded in Pn , forming Hilbert schemes, taking quotient stacks of the appropriate loci of normally embedded varieties by the action of PGL(n + 1), and taking the union

4

D. Abramovich and B. Hassett

of all the resulting stacks over all types of embeddings. Then one tries to carve out pieces of this “mother of all moduli spaces” (of varieties) which are bounded, and if lucky, proper, by ﬁxing appropriate numerical invariants. Miles Reid has been advocating over the years that, while most people insist on embedding projective varieties in projective space, the varieties really beg to be embedded in weighted projective spaces, denoted P(ρ) below. This is simply because graded rings are in general not generated in one degree; see [Rei78], [Rei97], and especially [Rei02]. For our stacks this point is absolutely essential, and in fact our stacks (as well as varieties) are embedded in the appropriate weighted projective stacks, denoted P(ρ) below. 1.5. Moduli of stacks: this paper. In section 2 we consider weighted projective stacks and their substacks. Following [OS03] substacks are parametrized by a Hilbert scheme. All substacks of P(ρ) are cyclotomic, in particular tame, and in analogy with the scheme case, we characterize in Proposition 2.4.2 pairs (X , L) consisting of a cyclotomic stack with a polarizing line bundle - one aﬀording an embedding - in cohomological terms. Similar results are obtained in [RT09], though the emphasis is diﬀerent. Ross and Thomas embed smooth orbifolds in weighted projective stacks to study orbifold constant scalar curvature K¨ahler metrics. They develop a notion of Geometric Invariant Theory stability for polarized orbifolds and use the relation between stability and moment maps to formulate obstructions to existence of these metrics. In section 3 we use these results to construct an algebraic stack StaL parametrizing pairs (X , L) of a proper stack with polarizing line bundle L (see Theorem 3.1.4). This involves studying the eﬀect of changing the range of weights on the moduli of normally embedded substacks (see Proposition 3.2.4). In section 4 we introduce orbispaces, discuss the diﬀerence between polarizing line bundle and polarization, and construct algebraic stacks OrbL and Orbλ parametrizing orbispaces with polarizing line bundles and polarized orbispaces, respectively (see Proposition 4.2.1). We further discuss the canonical polarization, and the algebraic stack Orbω of canonically polarized orbispaces, in Theorem 4.4.4. This ends the general discussion of moduli of stacks, and brings us to the questions related to Koll´ar families. We begin Section 5 by discussing Koll´ar families of Q-line bundles (see Deﬁnition 5.2.1). We then deﬁne uniformized twisted varieties in Deﬁnition 5.3.4. An important point here is that the conditions on ﬂat families are entirely on geometric ﬁbers. Finally we relate the two notions and prove an equivalence of categories between Koll´ ar families (X → B, F ) of Q-line bundles and families of uniformized twisted varieties (see Theorem 5.3.6). In particular this means that we have an algebraic stack KL of Koll´ar families with a polarizing Q-line bundle, which is at the same time the stack of twisted varieties with polarizing line bundles. The rigidiﬁcation Kλ of KL is naturally the stack of polarized twisted varieties. Passing to the locally closed substack where the initial homology group of the relative dualizing complex is invertible and polarizing, we obtain the substack Kω of canonically polarized twisted varieties.

5

Stable varieties

We end the paper by applying the previous discussion to moduli of stable varieties in Section 6. We deﬁne the moduli functor in terms of families of twisted stable varieties and equivalently Koll´ ar families of stable varieties. We show that this is open in Kω (see Proposition A.1.1) and discuss what is known about properness in section 6.1. It may be of interest to develop a treatment of the steps of the minimal model program, and not only the end product ProjR(X), using stacks. It would require at least an understanding of positivity properties for the birational “contraction” X → X from a twisted variety to its coarse moduli space, where we expect the notion of age, introduced in [IR96], to be salient. This however goes beyond the scope of the present paper. 1.6. Acknowledgments. We are grateful to Valery Alexeev, Kai Behrend, Tim Cochran, Alessio Corti, Barbara Fantechi, J´anos Koll´ar, S´andor Kov´acs, Martin Olsson, Miles Reid, Julius Ross, Richard Thomas, Eckart Viehweg, and Angelo Vistoli for helpful conversations about these questions. We appreciate the hospitality of the Mathematisches Forschungsinstitut Oberwolfach and the Mathematical Sciences Research Institute in Berkeley, California. 1.7. Conventions. Through most of this paper we work over Z. When we refer to schemes of ﬁnite type, we mean schemes of ﬁnite type over Z. The main exceptions are Section 6 and the Appendix, where we assume characteristic zero. We leave it to the interested reader to supply the incantations needed to extend our results to an arbitrary base scheme. In characteristic zero, we freely use the established literature for DeligneMumford stacks. In positive and mixed characteristics, we rely on the recent paper [AOV08a], which develops a new notion of ‘tame stacks’ with nice properties, e.g., they admit coarse moduli spaces that behave well under base extension. 1.8. Notation. P(ρ) P(ρ) StaL OrbL Orbλ Orbω KL Kλ Kω ✷cm ✷gor ✷can ✷slc ✷gor-slc ✷can

Weighted projective space of Gm -representation ρ Weighted projective stack of Gm -representation ρ Stack of stacks with polarizing line bundle Stack of orbispaces with polarizing line bundle – with polarization – with canonical polarization Stack of twisted varieties with polarizing line bundle – with polarization – with canonical polarization Index indicating Cohen-Macaulay ﬁbers – Gorenstein ﬁbers – canonical singularities (in the coarse ﬁbers) – semilog canonical singularities ﬁbers – Gorenstein s.l.c. singularities (in the twisted ﬁbers) indicating canonical model of something

6

D. Abramovich and B. Hassett

2. Cyclotomic stacks and weighted projective stacks 2.1. Weighted projective stacks. Deﬁnition 2.1.1. Fix a nondecreasing sequence of positive integer weights 0 < ρ1 ≤ . . . ≤ ρr and consider the associated linear action of Gm on aﬃne space ρ : Gm × Ar

→

Ar

ρ∗ (x1 , . . . , xr )

=

(tρ1 x1 , . . . , tρr xr ).

Recall that weighted projective space associated to the representation ρ is deﬁned as the quotient scheme P(ρ) = P(ρ1 , . . . , ρr )

=

(Ar 1 0) / Gm .

We deﬁne the weighted projective stack as the quotient stack 9 H r P(ρ) = P(ρ1 , . . . , ρr ) = (A 1 0) / Gm . Each weighted projective stack has the corresponding weighted projective space as its coarse moduli space. For each positive integer d we have P(dρ1 , . . . , dρr ) @ P(ρ1 , . . . , ρr ), however, the weighted projective stacks are not isomorphic unless d = 1. Indeed P(dρ1 , . . . , dρr ) → P(ρ1 , . . . , ρr ) is a gerbe banded by µd . Note also that a weighted projective stack is representable if and only if each weight ρi = 1; it contains a nonempty representable open subset if and only if gcd(ρ1 , . . . , ρr ) = 1. Deﬁnition 2.1.2. Let B be a scheme of ﬁnite type and V a locally-free OB module of rank r. Suppose that Gm acts OB -linearly on V with positive weights (ρ1 , . . . , ρr ). Let w1 < . . . < wm denote the distinct weights that occur, so we have a decomposition of OB -modules V with Write A := 0 ⊂ A, and

@ Vw1 ⊕ . . . ⊕ Vwm

ρ(v) = twi v, SpecB (Sym•B V

v ∈ Vwi .

) for the associated vector bundle, with its zero section ρ : Gm ×B A → A

for the associated group action. The weighted projective stack associated with V and ρ is the quotient stack P(V, ρ) := [ (A 1 0) / Gm ] → B. We will sometimes drop V from the notation when the meaning is unambiguous.

7

Stable varieties

Graded coherent OA -modules descend to coherent sheaves on P(V, ρ); thus for each integer w, the twist OA (w) yields an invertible sheaf OP(V,ρ) (w) on P(V, ρ). The canonical graded homomorphisms Vwi ⊗B OA → OA (wi ) induce

φwi : Vwi ⊗B OP(V,ρ) → OP(V,ρ) (wi ),

i = 1, . . . , m.

Note that φwi vanishes along the locus where the elements of Vwi are simultaneously zero, i.e., where all the weight-wi coordinates simultaneously vanish. The homomorphisms φwi , i = 1, . . . , m do not vanish simultaneously anywhere. The following lemma is well-known in the case of projective space. The proof for weighted-projective stacks is identical: Lemma 2.1.3. The stack P(V, ρ) is equivalent, as a category ﬁbered over the category of B-schemes, with the following category: (1) Objects over a scheme T consist of (a) a line bundle L over T , and (b) OT -linear homomorphisms φwi : Vwi ⊗B OT → Lwi ,

i = 1, . . . , m,

not vanishing simultaneously at any point of T . (2) Arrows consist of ﬁber diagrams LD ↓ TD

→ →

L ↓ T

compatible with the homomorphisms φwi and φDwi . As we shall see, certain stacks admitting “uniformizing line bundles” have representable morphisms into weighted projective stacks. 2.2. Closed substacks of weighted projective stacks. By descent theory, closed substacks X ⊂ P(ρ) = P(ρ1 , . . . , ρr ) correspond to closed subschemes X ×P(ρ) (Ar 1 0) ⊂ (Ar 1 0), equivariant under the action of Gm . Let CX denote the closure of this scheme in Ar , i.e., the cone over X . Each such subscheme over a ﬁeld k can be deﬁned by a graded ideal I ⊂ k[x1 , . . . , xr ].

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D. Abramovich and B. Hassett

The saturation of such an ideal is deﬁned as d

{f ∈ k[x1 , . . . , xr ] : mx1 , . . . , xr C f ⊂ I for some d D 0} and two graded ideals are equivalent if they have the same saturation. Equivalent graded ideals give identical ideal sheaves on Ar 1 0, hence they deﬁne the same equivariant subschemes of Ar 1 0 corresponding to the same substack of P(ρ). There is one crucial distinction from the standard theory of projective varieties: when the weights ρi = 1 for all i, every graded ideal is equivalent to an ideal generated by elements in a single degree. This can fail when the weights are not all equal. However, if we set N = lcm(ρ1 , . . . , ρr ) then every graded ideal is equivalent to one with generators in degrees n, n + 1, . . . , n + N − 1, for n suﬃciently large. (This can be be shown by regarding graded ideals as modules over the subring of k[x1 , . . . , xr ] generated by monomials of weights divisible by N .) An equivariant subscheme CX ⊂ Ar is speciﬁed by the induced quotient homomorphisms on the graded pieces of the coordinate rings k[x1 , . . . , xr ]d $ k[CX ]d ,

d = n, n + 1, . . . , n + N − 1.

In sheaf-theoretic terms, OP(ρ) -module quotients OP(ρ) $ OX are determined by the induced quotient of OP(ρ) -modules N −1 −1 π∗ [⊕N i=0 OP(ρ) (i)] → π∗ [⊕i=0 OX (i)],

or any other interval of twists of length N , where π : P(ρ) → P(ρ) is the coarse moduli morphism. Deﬁnition 2.2.1. Fix a representation (V, ρ) of Gm with isotypical components Vw1 , . . . , Vwm of positive weights w1 < · · · < wm . A closed substack X ⊂ P(ρ) is said to be normally embedded if (1) for each 1 ≤ j ≤ m, the homomorphism Vwj → H 0 (X , OX (wj )) is an isomorphism, (2) for each k ≥ w1 and i ≥ 1 we have H i (X , OX (k)) = 0, and (3) the natural homomorphism Sym V

−→ ⊕k≥0 H 0 (X , OX (k))

surjects onto the component H 0 (X , OX (k)) for each k ≥ w1 . We proceed, following Olsson–Starr [OS03] to classify substacks X ⊂ P(ρ) by a Hilbert scheme. The following deﬁnition is [OS03, Deﬁnition 5.1].

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Stable varieties

Deﬁnition 2.2.2. A locally free sheaf E on a stack Y is generating if, for each quasi-coherent OY -module F, the natural homomorphism θ(F) : π ∗ π∗ HomY (E, F) ⊗Y E → F is surjective. We note that

N −1 E = ⊕i=0 OP(ρ) (−i)

is a generating sheaf for P(ρ) where N = lcm(w1 , . . . , wm ). The main insight of [OS03] §6 is to reduce the study of OP(ρ) -quotients of F to an analysis of OP(ρ) -quotients of π∗ HomP(ρ) (E, F). The following is a direct application of [OS03, Theorem 1.5] to weighted projective stacks. Theorem 2.2.3. Let ρ be an action of Gm on Ar with positive weights and P(ρ) the resulting weighted projective stack. For each scheme T and stack Y, write YT = Y × T . Consider the functor Hilb : Z-schemes → Sets with

@ HilbP(ρ) (T ) =

Isomorphism classes of OP(ρ)T -linear quotients OP(ρ)T $ Q, with Q ﬂat over T

@ =

Closed substacks X ⊂ P(ρ)T over T, with X ﬂat over T

<

< .

Then HilbP(ρ) is represented by a scheme (denoted also HilbP(ρ) ). Furthermore, for each function F : Z → Z, the subfunctor consisting of closed substacks X ⊂ P(ρ)T with Hilbert-Euler characteristic χ(Xt , OXt (m)) = F(m) for all geometric points t ∈ T is represented by a projective scheme HilbP(ρ),F which is a ﬁnite union of connected components of HilbP(ρ) . 2.3. Cyclotomic stacks and line bundles. Given an algebraic stack X , the inertia stack is an algebraic stack whose objects are pairs (ξ, σ), with ξ an object of X and σ ∈ Aut (ξ). The morphism IX → X given by (ξ, σ) ?→ ξ is representable, and IX is a group-scheme over X . If X is separated, the morphism IX → X is proper. Deﬁnition 2.3.1. A ﬂat separated algebraic stack f :X →B locally of ﬁnite presentation over a scheme B is said to be a cyclotomic stack if it has cyclotomic stabilizers, i.e., if each geometric ﬁber of IX → X is isomorphic to the ﬁnite group scheme µn for some n.

10

D. Abramovich and B. Hassett

Keeping X separated and with ﬁnite diagonal, let π : X → X denote the coarse-moduli-space morphism; it is proper and quasi-ﬁnite [KM97]. The resulting family of coarse moduli spaces is denoted f¯ : X → B. Example 2.3.2. Let Y → B be an algebraic space, ﬂat and separated over B. If Gm acts properly on Y over B with ﬁnite stabilizers then the quotient stack X := [Y /Gm ] → B is cyclotomic. Indeed, the stabilizer scheme Z = {(y, h) : h · y = y} ⊂ Y ×B Gm is equivariant with Gm -action on Y ×B Gm g · (y, h) = (gy, ghg −1 ) = (gy, h). On taking quotients, we obtain IX

=

[Z / Gm ] ,

which is a closed substack of [(Y ×B Gm ) / Gm ] = [X ×B Gm ] . Deﬁnition 2.3.3. We say that a stack X has index N if for each object ξ ∈ X (T ) over a scheme T , and each automorphism a ∈ Aut ξ we have aN = id, and if N is the minimal positive integer satisfying this condition. Lemma 2.3.4. Let

f :X →B

be a stack of ﬁnite presentation over a scheme B and having ﬁnite diagonal. There is a positive integer N such that X has index N . Proof. This is well known: we may assume X is of ﬁnite type over a ﬁeld k. Then the inertia stack IX is of ﬁnite type as well. The projection morphism IX → X is ﬁnite, with ﬁbers the automorphism group-schemes of the corresponding objects of X . The degree is therefore bounded. We note that cyclotomic stacks are tame [AOV08a, Theorem 3.2 (b)]. The following lemma is a special case of [AV02, Lemma 2.2.3] in the Deligne–Mumford case, and [AOV08a, Theorem 3.2 (d)] in general: Lemma 2.3.5. Consider a cyclotomic stack f :X →B and a geometric closed point ξ : Spec K → X with stabilizer µn . Let ξ¯ : Spec K → ¯ the stack X be the corresponding point. Then, in a suitable ´etale neighborhood of ξ, X is a quotient of an aﬃne scheme by an action of µn .

11

Stable varieties

The following is a special case of [AV02, Lemma 2.2.3] in the tame Deligne– Mumford case, and [AOV08a, Corollary 3.3] in general. Lemma 2.3.6. Let f : X → B be a cyclotomic stack (assumed by deﬁnition ﬂat). (1) The coarse moduli space f¯ : X → B is ﬂat. (2) Formation of coarse moduli spaces commutes with base extension, i.e., for any morphism of schemes X D → X, the coarse moduli space of X ×X X D is X D . In particular, the geometric ﬁbers of f¯ are coarse moduli spaces of the corresponding ﬁbers of f . Lemma 2.3.7. Let X → B be a cyclotomic stack of index N and L an invertible sheaf on X . Then there is an invertible sheaf M on X and an isomorphism LN @ π ∗ M. Proof. Each automorphism group acts trivially on the ﬁbers of LN and hence trivially on the sheaf LN . Write X locally near a geometric point ξ¯ of X as [U/µr ] with r|N ; assume L is trivial on U . Then the total space of LN is the quotient [(U × A1 )/µr ], but the action on the factor A1 is trivial, and the total space is [U/µr ] × A1 = X × A1 . Its coarse moduli space in this neighborhood is X × A1 . It follows that M = π∗ LN is an invertible sheaf on the coarse moduli space and π ∗ M → L is an isomorphism. See also [AOV08b, Lemma 2.9]. Deﬁnition 2.3.8. Let L be an invertible sheaf over a stack X . The stack PL := SpecX

7

⊕i∈Z Li

G

→ X

is called the principal bundle of L. The grading induces a Gm -action on PL over X , which gives PL the structure of a Gm -principal bundle over X . This deﬁnition requires the spectrum of a ﬁnitely-generated quasi-coherent sheaf of algebras, whose existence is guaranteed by [LMB00] §14.2. We shall require the following, see [AV02, Lemma 4.4.3] or [LMB00]. Lemma 2.3.9. Let X and Y be algebraic stacks. Let g : X → Y be a morphism. The following are equivalent: (1) g is representable; (2) for any algebraically closed ﬁeld K and geometric point ξ : Spec(K) → X , the natural homomorphism of group schemes Aut ξ → Aut g(ξ) is a monomorphism.

12

D. Abramovich and B. Hassett

Proof. Recall that an algebraic space is a stack with trivial stabilizers. Also note that if V is an algebraic space and φ : V → Y a morphism, then the automorphisms of a point ξ of X ×Y V are precisely the kernel of the map Aut φ(ξ) → Aut g(φ(ξ)). So if the condition on automorphisms holds then X ×Y V has trivial stabilizers and therefore g is representable. Conversely, if g is representable and V → Y is surjective then X ×Y V is representable and X ×Y V → X surjective. So since X ×Y V has trivial stabilizers we have Aut φ(ξ) → Aut g(φ(ξ)) injective for all ξ. We can now state a result describing generating sheaves coming from powers of a line bundle. Proposition 2.3.10. Let X → B be a cyclotomic stack of index N and L an invertible sheaf on X . The following conditions are equivalent. (1) PL is representable; (2) the classifying morphism X → BGm associated to PL is representable; −1 ⊗−i is a generating sheaf for X ; (3) ⊕N i=0 L

(4) for each geometric point ξ : Spec K → X the action of Aut ξ on the ﬁber of L is eﬀective. Deﬁnition 2.3.11. When any of the conditions in the proposition is satisﬁed, X → B is said to be uniformized by L, and L is a uniformizing line bundle for X → B. Proof of Proposition. We ﬁrst show that the ﬁrst two conditions are equivalent. The representable morphism Spec Z → BGm is the universal principal bundle, thus PL = X ×BGm Spec Z. Clearly if X → BGm is representable, so is PL . Conversely, if PL is representable, the automorphism group of any geometric object Spec K → X of X acts on the Gm torsor Spec K ×X PL with trivial stabilizers; in particular, it acts eﬀectively. This means that the morphism X → BGm induces a monomorphism on automorphism groups, and by Lemma 2.3.9 it is representable. Recall from [OS03], Proposition 5.2 that a sheaf F is a generating sheaf for X if and only if for every geometric point ξ : Spec(K) → X and every irreducible representation V of Aut (ξ), the representation V occurs in Fξ . The third and fourth conditions are therefore equivalent. We now show that the third condition is equivalent to the ﬁrst. Assume the third condition, and ﬁx a geometric point ξ : Spec(K) → X . Then N −1 i every irreducible representation of µr occurs in the ﬁber ⊕i=0 Lξ . This implies that the character Lξ is a generator of Z/rZ, and it clearly acts freely on the

13

Stable varieties L

principal bundle of L. Conversely, if X → BGm is representable, then µr = Aut (ξ) injects into the automorphism group Gm of Lξ , therefore the character of µr in Lξ is a generator the dual group Z/rZ, and therefore all characters occur in r−1 −i −1 −i ⊕i=0 Lξ ⊂ ⊕N i=0 Lξ . 2.4. Embedding of a cyclotomic stack in a weighted projective bundle. Deﬁnition 2.4.1. Let f : X → B be a proper cyclotomic stack, with moduli space f¯ : X → B. A polarizing line bundle L on X is a uniformizing line bundle, such that there is an f¯-ample invertible sheaf M on X , an integer N and an isomorphism LN @ π ∗ M. We have the following analogue of the standard properties of ample bundles: Proposition 2.4.2. Let X → B be a proper family of stacks over a base of ﬁnite type, and let L be a line bundle. The following conditions are equivalent: (1) The stack X is cyclotomic, and L is a polarizing line bundle. (2) For every coherent sheaf F on X there exist integers n0 and N such that for any n ≥ n0 the sheaf homomorphism n+N .−1

7 G H 0 X , Lj ⊗ F ⊗ L−j

→ F

j=n

is surjective. (3) The stack X is cyclotomic, uniformized by L, and for every coherent sheaf F on X there exists an integer n0 such that for any n ≥ n0 and any i ≥ 1 we have H i (X , Ln ⊗ F) = 0. Proof. Assume the ﬁrst condition and let F be a coherent sheaf. Let N be the index of X . Fix k such that Lk = π ∗ M with M ample and consider the sheaves Gi = π∗ (Li ⊗ F), i = 0, . . . N − 1 on M . There exists m0 such that for all m ≥ m0 the sheaf homomorphisms H 0 (X , M m ⊗ Gi ) ⊗ OX → Gi are surjective. Note 7that M m ⊗ GGi @7 π∗ (Lmk+iG⊗ F). Also by Proposition 2.3.10 we have that N −1 −i −1 i → F is surjective. The second condition follows. π ∗ π∗ ⊕N i=0 L ⊗ F ⊗ ⊕i=0 L The reverse implication also follows: L is uniformizing since the sheaf ⊕0i=−(N −1) Li is necessarily generating by Proposition 2.3.10, and M is ample by Serre’s criterion. Since X is tame we have that π∗ is exact on coherent sheaves. Writing n = mk + j we have H i (X , Ln ⊗ F) = H i (X, π∗ (Ln ⊗ F)) = H i (X, M m ⊗ π∗ (Li ⊗ F)). The equivalence of the ﬁrst two conditions with the third follows from Serre’s criterion for ampleness applied to M . Proposition 2.4.3. Let f : X → B be as above, and L a polarizing line bundle. There are positive integers n < m such that

14

D. Abramovich and B. Hassett

(1) for every geometric point s ∈ B with residue ﬁeld K, we have H i (Xs , Lj ) = 0 ∀ i > 0, j ≥ n; (2) the sheaf f∗ Lj is locally free ∀j ≥ n; (3) for every geometric point s ∈ B, the space m .

H 0 (Xs , Lj )

j=n

generates the algebra K

⊕

.

H 0 (Xs , Lj ).

j≥n

Proof. Part (1) is a special case of part 3 of Proposition 2.4.2. Part (2) follows from (1) by Grothendieck’s “cohomology and base change” applied directly to the f -ﬂat sheaves Lj since H 1 (Xs , Lj ) = 0. Alternatively, consider the sheaves π∗ Lj . These sheaves are ﬂat over B since X is tame: since Lj is ﬂat we have that for any OB -ideal I the induced homomorphism I ⊗OB Lj → Lj is injective. Since X is tame we have π∗ exact, therefore I ⊗OB π∗ Lj → π∗ Lj is injective, as needed. Since H 1 (Xs , π∗ Lj ) = 0 we can apply “cohomology and base change” to X → B. For (3), we ﬁrst note that a range n ≤ j ≤ m as required exists for any point s of B: for each i = 0, . . . , N − 1 there exists an l such that H 0 (X , π ∗ M l ⊗ Li ) ⊗ H 0 (X , π ∗ M r ) → H 0 (X , π ∗ M l+r ⊗ Li ) is surjective for all r. We may choose a single l that works for all i = 0, . . . , N − 1 and take n = lN . If at the same time we choose n so that part (1) holds, then elements of H 0 (Xs , Lj ) in the given ﬁnite range lift to a neighborhood of s. Also the algebra H 0 (X, ⊕l≥0 M l ) is ﬁnitely generated and for each j, H 0 (X, (π∗ Lj ) ⊕ M l ) is a ﬁnite module over the algebra. So a given range for Xs works for a neighborhood of s. Since B is Noetherian, ﬁnitely many such neighborhoods cover B and the maximal choice of m works over all of B. Corollary 2.4.4. Let n, m be as in Proposition 2.4.3. Consider the locally free sheaves Wj = f∗ Lj . Then we have a closed embedding X n→ P

m 7.

G Wj .

j=n

Proof. Write Rn,m := SymB

m 7. j=n

G Wj ,

15

Stable varieties

and denote A := SpecB Rn,m . We have a Gm -equivariant map PL → A, and since LN = M is ample, the image is disjoint from the zero section 0 ⊂ A. In order to show that the morphism of quotient stacks H7 G 9 [PL /Gm ] = X → A 1 0 /Gm is an embedding, it suﬃces to show that the morphism PL → A is an embedding. Let . Rn = OB ⊕ Wj . j≥n

We have a surjective algebra homomorphism Rn,m → Rn , implying that SpecB Rn → A is a closed embedding. It > suﬃces to show that PL → SpecB Rn is an embedding. Next, let RL = R0 = j≥0 Wj . The morphism SpecB RL → SpecB Rn induced by the natural inclusion Rn ⊂ RL is ﬁnite and birational, having its conductor supported along the zero section. It therefore suﬃces to show that the morphism PL → SpecB RL is an embedding. We now use the ﬁnite morphism PL → PM , given by the ﬁnite ring extension ⊕l∈Z M l n→ ⊕j∈Z π∗ Lj . Since M is ample, we have an open embedding PM n→ SpecB RM . Taking the normalization PˆL of SpecB RM in the structure sheaf of PL we get an embedding PL → PˆL . Note that PˆL is aﬃne over B. Now sections over aﬃnes in B of the structure sheaf of PˆL restrict to sections of π∗ Lj , in other words they come from global sections of π∗ Lj . It follows that PˆL = SpecB ⊕j∈Z f∗ Lj , and since these vanish for j < 0 we get PˆL = SpecB RL .

3. Moduli of stacks with polarizing line bundles 3.1. The category StaL . We are now poised to deﬁne a moduli stack. In order to avoid cumbersome terminology and convoluted statements, we will deﬁne the objects of the stack over schemes of ﬁnite type. The extension to arbitrary schemes is standard but less than illuminating. Deﬁnition 3.1.1. We deﬁne a category StaL , ﬁbered in groupoids over the category of schemes, as follows: (1) An object StaL (B) over a scheme of ﬁnite type consists of a proper family X → B of cyclotomic stacks, with polarizing line bundle L.

16

D. Abramovich and B. Hassett

(2) An arrow from (X → B, L) to (X1 → B1 , L1 ) consists of a ﬁber diagram X # B

φ

! X1 # ! B1

along with an isomorphism α : L → φ∗ L1 . Remark 3.1.2. As deﬁned, this is really a 2-category, because arrows of stacks in general have automorphisms, but because L is polarizing it is easy to see (see, e.g. [AGV08], §3.3.2) that it is equivalent to the associated category, whose morphisms are isomorphism classes of 1-morphisms in the 2-category. We can realize this category directly using the principal bundle PL as follows: an object of StaL (B) consists of a Gm -scheme P → B, such that X := [P/Gm ] is a proper cyclotomic stack polarized by the line bundle associated to P; an arrow is a Gm -equivariant ﬁber diagram ! P1 P # B

# ! B1 .

The fact that StaL parametrizes schemes with extra structure can be used to show that it is a stack. We will however show that it is an algebraic stack via a diﬀerent route - as suggested by the theorem below. It is also worth noting that StaL is highly non-separated and far from ﬁnite type. Remark 3.1.3. Note that the group Gm acts by automorphisms on every object. Theorem 3.1.4. The category StaL is an algebraic stack, locally of ﬁnite type over Z. In fact, StaL has an open covering by TQα → StaL , where (1) each Qα = [Hα /Gα ] is a global quotient, (2) Hα ⊂ HilbP(ρα ) is a quasi-projective open subscheme (described explicitly below), and (3) Gα = Aut ρα (V ), where V is the representation space of ρα . This theorem is a direct generalization of its well-known analogue, due to Grothendieck, in the context of moduli of projective schemes endowed with an ample sheaf. The proof encompasses the rest of this section. 3.2. Normally embeddable stacks. Before we start the proof in earnest, we give some preliminary results. Deﬁnition 3.2.1. A stack X uniformized by L is normally embeddable in P(ρ) if there exists a normal embedding X ⊂ P(ρ) such that OX (1) @ L.

17

Stable varieties

Proposition 3.2.2. Fix a function F : Z → Z and two positive integers n < m. For n ≤ i ≤ m ﬁx free modules Vi of rank F(i) and consider the natural representation ρ of Gm on V = ⊕m i=n Vi . (1) There is an open subscheme H ⊂ HilbP(ρ),F parametrizing normally embedded substacks. L (2) The subcategory StaL F (ρ) ⊂ Sta of objects with ﬁbers normally embeddable in P(ρ) and Hilbert–Euler characteristic χ(X , Lm ) = F(m) satisﬁes

StaL F (ρ) @ [H/G], with G = Aut ρ (V ). Remark 3.2.3. Note that G is the group of automorphisms of the free module representation rather than automorphisms of its projectivization. This accounts for the fact that Gm acts on all objects of the quotient [H/G]. Proof of Proposition. For (1), note that each of the conditions in Deﬁnition 2.2.1 is an open condition, by the theorem on cohomology and base change. We prove (2). The data of a map to the quotient stack B → [H/G] is equivalent to a family Y → Q → B where Q is a principal G-bundle over B and Y ⊂ P(ρ)×Q is a family of normally embedded substacks. Recall (see Deﬁnition 2.3.8) the principal Gm -bundle POY (1) → Y associated to OY (1). The free action of G lifts canonically to OY (1) and to its principal bundle, and commutes with the Gm -action on the line bundle by scaling. To show that Y → Q descends to B, consider the quotient POY (1) /G. This retains a Gm -action, with ﬁnite stabilizers, whose quotient is denoted X . Let L denote the line bundle associated to the principal Gm -bundle POY (1) /G → X . While we have scrupulously avoided the subtle procedure of taking the quotient of a stack under a free group action, we can identify X = Y/G and L = OY (1)/G, namely X parametrizes principal Gbundles carrying an object of Y and similarly for L. A diagram summarizing the objects in this construction, where the parallelograms are cartesian squares, is as follows: # # ! O(1) × H ! O(1) × Q ! OY (1) " POY (1) " && && && && && # # # # ' "# ! P(ρ) × Q ! P(ρ) × H PL % Y " %%% """ %%% """ %%% """ %%' # ""& # # !H Q X "" """ """ """ ""& # # ! [H/G]. B Now X → B is a cyclotomic stack polarized by L, with geometric ﬁbers embeddable in P(ρ): indeed the geometric ﬁbers of X → B are isomorphic to those of Y → Q,

18

D. Abramovich and B. Hassett

which are cyclotomic, polarized by OY (1), and embedded in P(ρ). The whole construction works over arrows B D → B → [H/G], and therefore gives a functor [H/G] → StaL F (ρ). It is crucial in this construction that we work with the quotient of H by the linear group G and not its projective version - otherwise we would not have a functorial construction of the line bundle L. We now consider the opposite direction. Let X → B be a cyclotomic stack j polarized by L. Consider the locally-free sheaf ⊕m n Wj , with Wj = f∗ L . Each ﬁber of W is isomorphic as Gm -space to a ﬁxed free module V with representation ρ. By the embeddability assumption and Corollary 2.4.4, we have an embedding X n→ P(W ). Consider the principal G-bundle Q = IsomGm (W, VB ). We have a canonical isomorphism W ×B Q @ VB , giving an isomorphism P(W ) ×B Q @ P(ρ) × Q. Write Y = X ×B Q. We have a canonical induced embedding Y n→ P(ρ) × Q. This is a normal embedding by deﬁnition, and it is clearly G-equivariant. We thus obtain an object of [H/G] as required. Again the fact that the construction is canonical gives a construction for arrows in B D → B → StaL F (ρ). The proof that the two functors are inverse to each other is standard and left to the reader. m.

We now consider what happens when we change the range of integers n ≤ i ≤

Proposition 3.2.4. Fix again a function F : Z → Z and free modules Vi of rank F(i). For positive integers n < m we again consider the natural representation ρn,m of Gm on Vn,m = ⊕m i=n Vi . Then L (1) StaL F (ρn,m ) → StaF (ρn,m+1 ) is an open embedding.

(2) For any n < m there exists a canonical open embedding L StaL F (ρn,m ) n→ StaF (ρn+1,m+n ).

(3) For any n1 < m1 and n2 < m2 , there exist an integer m ≥ max(m1 , m2 ) and canonical open embeddings L StaL F (ρn1 ,m1 ) n→ StaF (ρn,m ) L StaL F (ρn2 ,m2 ) n→ StaF (ρn,m )

with n = max(n1 , n2 ). Proof of proposition. For (1), it is clear that we have an inclusion StaL F (ρn,m ) n→ StaL (ρ ). Given a family of graded rings generated in degrees n, . . . , m + 1, F n,m+1 it is an open condition on the base that they are generated in degrees n, . . . , m, as required. L We prove (2). We have an open embedding StaL F (ρn,m ) n→ StaF (ρn,m+n ) by L (1). In StaL F (ρn,m+n ) we have an open substack StaF (ρ(n),n+1,m+n ) of objects

19

Stable varieties

whose projection to the factors in degrees n + 1, . . . , n + m is still an embedding. L This also has an open embedding StaL F (ρ(n),n+1,m+n ) n→ StaF (ρn+1,m+n ) as it i n corresponds to objects such that H (X , L ) = 0 for i > 0. " StaL F (ρn,m )

#

! StaL (ρn,m+n ) F ( "! ) "# StaL F (ρ(n),n+1,m+n )

! StaL (ρn+1,m+n ) F

Now consider a graded ring R = ⊕Ri . Suppose R is generated in degrees n, . . . m over R0 . Then the truncation ∞ .

R D = R0 ⊕

Ri

i=n+1

is generated in degrees n + 1, . . . , n + m. Indeed if g1 · · · gk is a product of homogeneous terms in degrees between n and m, of total degree > n + m, and is of minimal degree requiring a term g1 of degree n as a product of terms in the range n, . . . , m + n, then the other terms g2 , . . . , gk are of degree between n + 1 and m, so we may replace g1 by g1D = g1 gk and write g = g1D g2 · · · gk−1 as a product of homogeneous terms in the range n + 1, . . . , n + m. This implies that the truncation of the ring of an object in StaL F (ρn,m+n ) coming from the open substack StaL (ρ ) is still normally generated. This F n,m L L means that the embedding StaF (ρn,m ) ⊂ StaF (ρn,m+n ) factors through the stack L StaL F (ρ(n),n+1,m+n ). This induces the required open embedding StaF (ρn,m ) n→ L StaF (ρn+1,m+n ), as required. Part (3) follows by iterating the construction of part (2). 3.3. Proof of Theorem 3.1.4. Given an object f : X → B in StaL (B), there is a maximal open and closed subset B0 ⊂ B so that f0 : X ×B B0 → B0 is an object in StaL F (B0 ); the formation of B0 commutes with arbitrary base change. Thus we can decompose K StaL StaL = F, F

StaL F

where is the open and closed ﬁbered subcategory parametrizing cyclotomic stacks polarized by a line bundle with Hilbert-Euler characteristic F : Z → Z. We can thus ﬁx F and focus on StaL F. Proposition 3.2.4 gives for each pair of integers n < m a ﬁbered subcategory L StaL F (ρn,m ) ⊂ StaF . This ﬁbered subcategory is open: ﬁrst note that the condition that the higher cohomologies of Li vanish for n ≤ i ≤ m is open by cohomology and base change. Second, the condition that the ring be normally generated in these degrees (including vanishing of cohomology in higher degrees) is open as well. These open embeddings are compatible with the embeddings in Proposition L 3.2.4. It follows that ∪StaL F (ρn,m ) ⊂ StaF is an open subcategory. But by CorolL lary 2.4.4 every geometric point of StaL F is in ∪StaF (ρn,m ), which implies that

20

D. Abramovich and B. Hassett

L L ∪StaL F (ρn,m ) = StaF as ﬁbered categories. Since ∪StaF (ρn,m ) is a direct limit of open subcategories which are algebraic stacks, it is also an algebraic stack.

4. Moduli of polarized orbispaces 4.1. Orbispaces. Deﬁnition 4.1.1. An orbispace X is a separated stack with ﬁnite diagonal, of ﬁnite type over a ﬁeld, equidimensional, geometrically connected and reduced, and admitting a dense open U ⊂ X where U is an algebraic space. The assumption that X be connected is made mainly for convenience. Given two ﬂat families of orbispaces Xi → Bi there exists a natural notion of a 1-morphism of families between them, namely a cartesian square X1 ↓ B1

→ →

X2 ↓ B2 .

There is a natural notion of 2-morphism, making families of orbispaces into a 2category. However by [AV00], Lemma 4.2.3, such 2-morphisms are unique when they exist. It follows that this 2-category is equivalent to the associated category, where morphisms consist of isomorphism classes of 1-morphisms. We call this the category of families of orbispaces. As an example, we have that the weighted projective stack P(ρ1 , . . . , ρr ) is an orbispace if and only if gcd(ρ1 , . . . , ρr ) = 1. We can consider cyclotomic orbispaces with uniformizing line bundle. Adding the conditions in StaL we can deﬁne another category - the subcategory OrbL ⊂ StaL of proper cyclotomic orbispaces with polarizing line bundles. We have Proposition 4.1.2. The subcategory OrbL ⊂ StaL forms an open substack. Proof. Let X → B be a family of cyclotomic stacks polarized by L. The locus where the ﬁbers are reduced and connected is open in B: see [Gro66, 12.2.1], for “no embedded points” and then cohomology and base change for “connected”. Since the inertia stack is ﬁnite over X , the same holds for the locus where the ﬁbers have a dense open with trivial inertia. 4.2. Polarizations. There is an important distinction to be made between a polarizing line bundle and a polarization - the diﬀerence between the data of a line bundle and an element of the Picard group. Let (Xi → Bi , Li ) be two families with polarizing line bundles Li ; then a morphism comes from a 1-morphism fX : X1 → X2 sitting in a cartesian diagram as above, together with an isomorphism α : L1 → fX∗ L2 . But a morphism of polarizations should ignore the Gm -action on the line bundles. The issue is treated extensively in the literature. A procedure for removing the redundant action, called

21

Stable varieties

rigidiﬁcation, is treated in [ACV03], [Rom05], [AGV08], [AOV08a, Appendix A]. This is foreshadowed by the appendix in [Art74]. Going back to our families with polarizing line bundles, the hypothesis that there is at least one point in each ﬁber where inertia is trivial implies that Gm is a subgroup of the center of Aut (X , L) for any object (when the generic stabilizer is nontrivial, Gm need not act eﬀectively). This allows us to rigidify the stack OrbL along the Gm -action. Following the notation of [Rom05, AGV08, AOV08a] (but diﬀerent from [ACV03]) we thus have an algebraic stack Orbλ = OrbL! Gm . An object (X → B, λ) of Orbλ (B) is a polarized family of cyclotomic stacks over B. These will be described below. Recall that we have a presentation StaL = ∪[Hα /Gα ], with Hα a subscheme of the Hilbert scheme of the weighted projective stack P(ρα ). Whenever P(ρα ) is an orbispace the group Gm embeds naturally in the center of the group Gα , and we can form the projective group PGα = Gα /Gm . We can extract from this a presentation of OrbL and of Orbλ as follows: Proposition 4.2.1. Let Hαorb ⊂ Hα be the open subscheme parametrizing embedded orbispaces. Then : (1) OrbL = α [Hαorb /Gα ]. : (2) Orbλ = α [Hαorb /PGα ]. This follows directly from the construction in any of [ACV03], [AGV08], or [AOV08a, Appendix A]. This construction induces a morphism of stacks OrbL → Orbλ , which gives OrbL the structure of a gerbe banded by Gm over Orbλ . The stack Orbλ can be described as follows: an object (X → B, λ) of Orbλ (B) is a family of cyclotomic stacks X → B, and λ given locally in the ´etale topology on B by a polarizing line bundle on X . Moreover, these ‘almost’ descend to a polarizing line bundle on X , i.e., on the overlaps these line bundles diﬀer by a line bundle coming from the base; this determines a section λ of Pic(X /B) called a polarization of X . The obstruction to existence of a line bundle on X is exactly the Brauer class in H´e2t (B, Gm ) of the Gm -gerbe B ×Orbλ OrbL → B. If the obstruction is trivial, then a polarizing line bundle L exists. In this case two pairs (X , L) and (X , LD ) represent the same polarization if and only if there is a line bundle M on B such that L @ LD ⊗ f ∗ M . 4.3. Comparison of polarizations. The construction of rigidiﬁcation involves stackiﬁcation of a pre-stack. In our case this is much simpler as we end up sheaﬁfying a pre-sheaf. The underlying fact is the following well-known result (cf. [Vie95, Lemma 1.19]):

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Proposition 4.3.1. Consider a proper family of stacks f : X → B and L, LD line bundles on X . Assume the ﬁbers of X → B are reduced and connected. There exists a locally closed subscheme B 0 ⊂ B and a line bundle M on B 0 , over which there exists an isomorphism L|B 0 → LD |B 0 ⊗ fB∗ 0 M , and is universal with respect to this property. By ‘universal’ we mean that if B 1 → B is a morphism and M1 a line bundle on B such that there exists an isomorphism L|B 1 → LD |B 1 ⊗ fB∗ 1 M1 , then B 1 → B factors uniquely through B 0 . The following is the outcome for polarizations. 1

Corollary 4.3.2. Consider a proper family of orbispaces X → B and λ, λD polarizations on X . There exists a locally closed subscheme B 0 ⊂ B, over which there exists an isomorphism λB 0 → λDB 0 , and is universal with respect to this property. Proof. Fix an ´etale covering C → B and line bundles L, LD on XC representing λC , λDC . By the proposition there is C 0 ⊂ C and a line bundle M on C 0 , over which there exists an isomorphism L|C 0 → LD |C 0 ⊗ fC∗ M , and is universal with respect to this property. Since it is universal, the two inverse images of C 0 in C ×B C coincide, therefore C 0 descends to B 0 ⊂ B, and the data L|C 0 → LD |C 0 ⊗ fC∗ M gives an isomorphism λB 0 → λDB 0 by deﬁnition. Corollary 4.3.3. Consider a proper family of orbispaces X → B, L a line bundle, λD a polarization on X . There exists a locally closed subscheme B 0 ⊂ B, over which there exists an isomorphism λB 0 → λDB 0 , with λ the polarization induced by L, and is universal with respect to this property. Proof. This is immediate from the previous corollary. 4.4. Moduli of canonically polarized orbispaces. Lemma 4.4.1. Let X → B be a proper family of orbispaces. There exist open subschemes Bgor ⊂ Bcm ⊂ B where the ﬁbers of X → B are Gorenstein and Cohen-Macaulay respectively. Proof. Recall the following fact [Gro66, 12.2.1]: Suppose X → B is a ﬂat morphism of ﬁnite type with pure dimensional ﬁbers and F is a coherent OX -module ﬂat over B. Then for each r ≥ 0 the locus {b ∈ B : F |Xb is Sr } is open in B. Take F = OX and r = dim(X/B) to see that the locus where the ﬁbers are Cohen-Macaulay is open. When the ﬁbers of X → B are Cohen-Macaulay, the shifted relative dualizing • complex ωX/B [−n] is a sheaf ωX/B that is invertible precisely on the open subset where the ﬁbers are Gorenstein (see [Con00], Theorem 3.5.1, Corollary 3.5.2, and the subsequent discussion).

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Stable varieties

It remains to check this analysis applies to the stack X . In characteristic 0, note that the dualizing sheaf is insensitive to ´etale localization [Con00, Th. 4.4.4 and p. 214], and thus descends canonically to X . In arbitrary characteristic one has to use smooth covers instead. However, it is a fundamental property of the dualizing complex that it behaves well under smooth morphisms, i.e., if g : Y → Z is smooth and Z → B is ﬂat then (up to shifts) • ⊗ ωY /Z . ωY• /B = g ∗ ωZ/B

Indeed, these formulas are fundamental tools in showing that the dualizing complex is well-deﬁned (see [Con00, pp. 29-30, Th. 2.8.1 and 3.5.1]). In particular, we can • deﬁne ωX /B via smooth covers. In the particular case at hand with X = [P/Gm ], the smooth cover is given by P and the descent datum is given by the Gm -equivariant structure on ωP . (Duality for Artin stacks is developed in [Nir08]. However we will only need the existence of a unique ωX .) The lemma implies: Proposition 4.4.2. There are open substacks Orbλgor ⊂ Orbλcm ⊂ Orbλ parametrizing Gorenstein and Cohen-Macaulay polarized orbispaces. For most applications to compactiﬁcations of moduli spaces these substacks should be suﬃcient (see Remark 6.1.6). However, in some situations more general singularities might be needed: Deﬁnition 4.4.3. A family X → B of orbispaces is canonically polarized if • the ﬁbers are pure-dimensional, satisfy Serre’s condition S2 , and are Gorenstein in codimension one; ◦ • • ωX /B , namely the component of the dualizing complex ωX /B in degree dim B − dim X, is locally free and polarizing. ◦ The canonical polarization on X → B is the polarization induced by ωX /B .

Theorem 4.4.4. There is a locally closed substack Orbω ⊂ Orbλ parametrizing canonically polarized orbispaces. Proof. Lemma 4.4.1 guarantees the conditions on the ﬁbers are open. The changeof-rings spectral sequence gives a homomorphism of sheaves ◦ ◦ ωX /B |Xb → ωXb .

This is injective because the relative dualizing sheaf is invertible and Xb is reduced; its cokernel is supported on a subset of Xb of codimension two. The dualizing sheaf ◦ ωX is automatically saturated (cf. [Rei87, 1.6]), so any injection from an invertible b sheaf that is an isomorphism in codimension one is an isomorphism. Thus on the ◦ open locus where ωX /B is invertible, its formation commutes with arbitrary base change. Applying Corollary 4.3.3, we obtain the theorem.

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ω λ We denote the intersection Orbω gor := Orb ∩Orbgor , the substack of canonically polarized Gorenstein orbispaces. Note that over Orbω , the universal polarization is represented by an invertible sheaf, namely, the relative dualizing sheaf of the universal family. This gives us a lifting L Orb "

Orbω

# ! Orbλ .

Indeed we can describe Orbω or Orbω gor directly in terms of line bundles as follows: an object of Orbω is a triple (X , L, φ) where X is an orbispace on which ◦ ◦ ωX /B is invertible, L a line bundle, and φ : L → ωX /B an isomorphism. Arrows ω are ﬁber diagrams as for Orb . But note that an arrow does not involve the choice ◦ of an arrow on ωX /B : such an arrow is canonically given as the unique arrow on ◦ ◦ ωX /B respecting the trace map Rn π∗ ωX /B → OB . The relative dualizing sheaf (and complex) does have Gm acting as automorphisms, but this automorphism group acts eﬀectively on the trace map.

5. Koll´ ar families and stacks 5.1. Reﬂexive sheaves, saturation and base change. Let X be a reduced scheme of ﬁnite type over a ﬁeld, having pure dimension d, satisfying Serre’s condition S2 . Lemma 5.1.1. (1) Let F be a coherent sheaf on X. Then the sheaf F ∗ := Hom(F, OX ) is S2 . (2) Suppose ψ : F → G is a morphism of S2 -sheaves which is an isomorphism on the complement of a closed subscheme of codimension ≥ 2. Then ψ is an isomorphism. (3) Let F be an S2 coherent sheaf on X. Then the morphism F → F ∗∗ is an isomorphism. Proof. The problems being local, we assume X is aﬃne. (1) Let φ be a section of Hom(F, OX ) over an open set U with codim(X 1U ) ≥ 2. Let f be a section of F on X and consider φ(fU ) ∈ H 0 (U, OU ). Since X is ˜ ) = g. S2 this extends uniquely to a regular function g on X. We deﬁne φ(f (2) Let g be a section of G. Let U be an open subset with codimension ≥ 2 complement on which ψ is an isomorphism. Set fU = ψ −1 (gU ), then fU is uniquely the restriction of a section f of F , and f = ψ −1 g. (3) Since F and F ∗∗ are S2 , the homomorphism F → F ∗∗ is an isomorphism.

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Deﬁnition 5.1.2. A coherent sheaf F on X is said to be reﬂexive if the morphism F → F ∗∗ is an isomorphism. Deﬁnition 5.1.3. Let F be a coherent sheaf on X, and n a positive integer. We deﬁne F [−n] = Hom(F ⊗n , OX ). and F [n] = Hom(F [−n] , OX ). Theorem 10 of [Kol95] implies Proposition 5.1.4. Let X → B be a ﬂat morphism of ﬁnite type with S2 ﬁbers of pure dimension d. Let U ⊂ X be an open subscheme, dense in each ﬁber. Let F be a coherent sheaf on X, locally free on U . If the formation of F [n] commutes with arbitrary base extension then F [n] is ﬂat over B. 5.2. Koll´ ar families of Q-line bundles. Deﬁnition 5.2.1.

(1) By a Koll´ ar family of Q-line bundles we mean

(a) f¯ : X → B a ﬂat family of equidimensional connected reduced schemes satisfying Serre’s condition S2 , (b) F a coherent sheaf on X, such that (i) for each ﬁber Xb , the restriction F |Xb is reﬂexive of rank 1; (ii) for every n, the formation of F [n] commutes with arbitrary base change; (iii) for each geometric point s of B there is an integer Ns K= 0 such that F [Ns ] |Xs is invertible. (2) A morphism from a Koll´ ar family (X → B, F ) to another (X1 → B1 , F1 ) consists of a cartesian diagram X # B

¯ φ

! X1 # ! B1

along with an isomorphism α ¯ : F → φ¯∗ F1 . (3) Parts (1) and (2) above deﬁne the objects and arrows of a category of Koll´ ar families of Q-line bundles, ﬁbered over the category Schk of k-schemes. It has an important open subcategory KL of Koll´ ar families of polarizing Q-line bundles, where X → B is proper and F [Ns ] |Xs is ample.

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Note that (1.b.i) and (1.b.ii) imply that F = F [1] . Furthermore, (1.b.i) and (1.b.iii) imply that F is invertible in codimension 1. A reﬂexive rank-one sheaf on a reduced one-dimensional scheme X is a fractional ideal J. Indeed, express ⊕r ⊕r F ∗ locally as a quotient OX → F → 0, which realizes F as a subsheaf of OX . ⊕r Choose a projection OX → OX such that the composed homomorphism F → OX has rank one at each generic point. This is injective since our scheme is reduced. The saturated powers F [n] equal J n , i.e., the powers of J as an ideal. In particular, J N is locally principal. The blow-up of X along J is ProjX ⊕n≥0 J n @ ProjX ⊕n≥0 J N n = X, so J is locally principal and F is invertible. One can show that the category of proper Koll´ ar families is an algebraic stack. We will show below that the subcategory KL is an algebraic stack by identifying it as an algebraic substack of OrbL . Koll´ar families were introduced in the canonical case F = ωX/B in [Kol90, 5.2] and [HK04, 2.11]. In [Hac04] it was shown that they admit, at least in the canonical case, a good deformation-obstruction theory. As we see below, this holds in complete generality. Indeed Hacking’s approach is via the associated stacks, as is ours. In [Kol08] Koll´ ar provides an approach without stacks. We note that the index Ns is bounded in suitable open sets: Lemma 5.2.2. Let (X → B, F ) be a Koll´ ar family of Q-line bundles, and s ∈ B a geometric point. Let Ns be an integer such that F [Ns ] |Xs is invertible. Then there is an open neighborhood U of s such that F [Ns ] |XU is invertible. Proof. Since the F [N ] are ﬂat and their formation commutes with base change, the assumption that F [Ns ] |Xs is invertible implies F [Ns ] is invertible in a neighborhood of Xs . Since X → B is of ﬁnite type the assertion follows. 5.3. Koll´ ar families and uniformized twisted varieties. Deﬁnition 5.3.1. Let (f¯ : X → B , F ) be a Koll´ar family of Q-line bundles. (1) The Gm -space of F is the X-scheme PF

Q 3 = SpecX ⊕j∈Z F [j] .

Note that PF → B is ﬂat by Proposition 5.1.4. (2) The associated stack is XF

=

H

PF

J

9 Gm .

This comes with the natural line bundle L = LF associated to the principal bundle PF → XF .

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We drop the index F and write X and P when no confusion is likely. We denote by π : X → X the resulting morphism. Proposition 5.3.2. (1) The family X → B is a family of cyclotomic orbispaces uniformized by L, with ﬁbers satisfying Serre’s condition S2 . (2) The morphism π : X → X makes X into the coarse moduli space of X . This morphism is an isomorphism on the open subset where F is invertible, the complement of which has codimension > 1 in each ﬁber. (3) For any integer a, we have π∗ (La ) = F [a] (in particular π∗ (La ) is reﬂexive). (4) This construction is functorial, that is, given a morphism of Koll´ ar families ¯ α ¯ ) from (X → B, F ) to (X1 → B1 , F1 ) we have canonically PF @ φ¯∗ PF1 (φ, and XF @ φ¯∗ XF1 . Proof. To verify that X → B is cyclotomic and uniformized by L, we just need to check that Gm acts on PF with ﬁnite stabilizers. Lemma 5.2.2 allows us to assume that F [N ] is locally free for some N > 0. We have natural homomorphisms ma : (F [N ] )a −→ F [N a] for each a ∈ Z. For each b ∈ B, the sheaf F is locally free on an open set Ub n→ Xb with codimension-two complement, so ma is an isomorphism over Ub , and hence over all Xb . It follows that (F [N ] )a = F [N a] . Consider the Gm -equivariant morphism Q 3 PF −→ SpecX ⊕a∈Z F [aN ] . Note that Gm acts on the target with stabilizer µN , which implies the stabilizers on PF are subgroups of µN . To complete part (1), we note that a ﬁber in PF over b ∈ B is S2 as the spectrum of an algebra with reﬂexive components over an S2 base. Also PF → X is smooth and surjective. Smoothness implies that the quotient stack X is S2 as well. Since the coarse moduli space is obtained by taking invariants (see the proof of Lemma 2.3.6), and since the invariant part of ⊕F [a] is F [0] = OX , we have that X is the coarse moduli space of X . Over the locus U where F is invertible, the scheme PF is a principal Gm -bundle, so XU = [ (PF )|U / Gm ] = U. Part (3) follows since La is the degree-a component of the graded OX -algebra ⊕L , whose spectrum is PF . The degree-a component of the graded OX algebra ⊕i∈Z F [i] is F [a] . Part (4) follows from the functoriality of Spec and of the formation of quotient stack. i

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Remark 5.3.3. If we assume for each a ∈ Z and b ∈ B that F [a] is Sr (in particular the ﬁbers of X are Sr ), it follows that the ﬁbers of X are Sr as well. Deﬁnition 5.3.4. A family of uniformized twisted varieties (X → B, L) is a ﬂat family of S2 cyclotomic orbispaces uniformized by L, such that the morphism π : X → X to the coarse moduli space is an isomorphism away from a subset of codimension > 1 in each ﬁber. We deﬁne 1-morphisms of families of uniformized twisted varieties as ﬁbered diagrams. Of course, there is a notion of 2-morphism making these into a 2category, but since these are orbispaces, a 2-isomorphism is unique when it exists [AV02, Lemma 4.2.3], so this 2-category is equivalent to the associated category, whose morphisms are isomorphism classes of 1-morphisms. Remark 5.3.5. We insist on π : X → X being an isomorphism in codimension one to obtain an equivalence with Koll´ ar families as in Theorem 5.3.6 below. To describe uniformized orbispaces in terms of sheaves on X in general, it is necessary to specify an algebra of sheaves, which can be quite delicate especially when X → X is branched along singular codimension-1 loci. For a subtle example see [Jar00, Deﬁnition 2.5] as well as [AJ03, §4.2-4.4], where an equivalence analogous to our Theorem 5.3.6 below is given. Theorem 5.3.6. The category of Koll´ ar families of Q-line bundles is equivalent to the category of uniformized twisted varieties via the base preserving functors (X → B, F )

?→

(XF → B, LF ),

?→

(X → B, π∗ L),

with XF = [PF / Gm ], and its inverse (X → B, L)

where X is the coarse moduli space of X . Proof. Proposition 5.3.2 gives the functor from the category of Koll´ ar families of Q-line bundles to the category of uniformized twisted varieties. We now give an inverse. Fix a uniformized twisted variety (X → B, L), with coarse moduli space π : X → X. Since the formation of π commutes with arbitrary base change (Lemma 2.3.6), the universal property of coarse moduli spaces guarantees that each ﬁber of X → B is reduced. If N is the index of X then, by Lemma 2.3.7, LN descends to an invertible sheaf on X, which coincides with π∗ LN . Each geometric point of X admits an ´etale neighborhood which is isomorphic to a quotient of an aﬃne S2 -scheme V by the action of µr . Locally, the coarse moduli space is the scheme-theoretic quotient V /µr . However, since µr is linearly reductive, any quotient of an S2 -scheme by µr is also S2 , because the invariants are a direct summand in the coordinate ring of V . Similarly the sheaves π∗ Lj are direct summands in the algebra of P, which is aﬃne over X, ﬂat over B with S2 ﬁbers. It follows that these sheaves are ﬂat over B, saturated, and their formation commutes with base change on B.

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Stable varieties

Corollary 5.3.7. The category KL is an algebraic stack, isomorphic to the open substack of OrbL where (X → B, L) are uniformized twisted varieties. Proof. Using Theorem 5.3.6 note that KL indeed parametrizes orbispaces with polarizing line bundles which are at the same time uniformized twisted varieties. This is open in OrbL since the condition of being S2 is open ([Gro66, 12.2.1]) and the condition on the ﬁbers of inertia having support in codimension > 1 is open by semicontinuity of ﬁber dimensions in proper morphisms. Deﬁnition 5.3.8. We deﬁne Kλ = KL! Gm , where Gm acts by scalars on L. We deﬁne Kω ⊂ Kλ as the locally closed subcategory corresponding to canonically ◦ polarized twisted varieties (f : X → B, λ) where λ is given by ωX /B . ω ω We deﬁne Kgor ⊂ K to be the open substack where the ﬁbers of f are CohenMacaulay. Since the dualizing sheaf is invertible–it is the polarizing line bundle–the ﬁbers are automatically Gorenstein. Note that Corollary 4.3.3 guarantees that the second condition is locally closed. Remark 5.3.9. We can interpret Kω in terms of Koll´ar families (f¯ : X → B, F ) ◦ ω with isomorphism F → ωX /B . The substack Kgor is characterized by (1) the ﬁbers of f are Cohen-Macaulay; [n]

(2) each power ωX/B is Cohen-Macaulay. Indeed, if a sheaf G on X is Cohen-Macaulay then π∗ G is also Cohen-Macaulay: locally write X as a quotient of an aﬃne scheme V by µr , and G corresponding to an equivariant module G. The complex computing local cohomology of G can be taken µr -equivariant, and the invariant subcomplex splits. Therefore, if the local cohomology vanishes, then the invariant part, which computes local cohomology of Gµr = π∗ G on X, vanishes as well. In general, the second condition is not a logical consequence of the ﬁrst. There are examples [Sin03, §6] of Cohen-Macaulay log canonical threefolds whose indexone covers are not Cohen-Macaulay. Remark 5.3.10. In characteristic zero, varieties with canonical (or even log terminal) singularities admit index-one covers with canonical singularities [Rei80], which are therefore rational [Elk81] and Cohen-Macaulay [KM98]. However, in positive characteristic there exist log terminal surface singularities with index-one covers that are not canonical [Kaw99b]. Generally, index-one covers present technical diﬃculties when the index of the singularity is divisible by the characteristic, especially in small characteristics. Our approach sidesteps these issues.

6. Semilog canonical singularities and compactiﬁcations From here on we assume our schemes are over a ﬁeld of characteristic 0.

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6.1. Properness results and questions. Recall that canonical singularities deform to canonical singularities [Kaw99a]. This fact and Proposition A.1.1 of the appendix allow us to formulate the following: ω Deﬁnition 6.1.1. We deﬁne Kcan ⊂ Kω as the open substack corresponding to canonically polarized twisted varieties where both X and X have canonical singularities. ω ω We deﬁne Kslc , Kgor-slc ⊂ Kω as the open substack corresponding to canonically polarized twisted varieties where X has semilog canonical (resp. Gorenstein semilog canonical) singularities. ω Objects of Kslc are called families of twisted stable varieties.

Theorem 5.3.6 induces equivalences of the categories above with the respective ω categories of Koll´ar families. Accordingly, we sometimes call Kslc the stack of Koll´ ar families of stable varieties, with the understanding that the Q-line bundle is the canonical polarization. We review what is known about properness of moduli spaces of stable varieties: ω ω Proposition 6.1.2. Each connected component of the closure of Kcan in Kslc is proper with projective coarse moduli space.

Proof. Most of this is contained in [Kar00], though our deﬁnition of families diﬀers from Karu’s. ω We ﬁrst check that the closure of Kcan satisﬁes the valuative criterion for properness. Let Δ be the spectrum of a discrete valuation ring, with special point s and generic point η. Let Xη be a canonically-polarized orbispace with Gorenstein canonical singularities over η. Let Xη denote its coarse moduli space. Apply resolution of singularities and semistable reduction [KKMSD73] to obtain • a ﬁnite branched cover ΔD → Δ with generic point η D and special point sD ; • a nonsingular variety Y and a ﬂat projective morphism φ : Y → ΔD , with Ys" reduced with simple normal crossings; • a birational morphism Yη" → Xη" . Consider the canonical model of Y relative to φ ψ : W → ΔD , which exists due to [BCHM10, Theorem 1.1]. It has the following properties: • KW/Δ" is Q-Cartier and ample relative to ψ; • W has canonical singularities, so every exceptional divisor has discrepancy ≥ 0.

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Stable varieties

As a consequence, we deduce Wη" @ Xη" and W is Cohen-Macaulay. Furthermore, every exceptional divisor in the special ﬁber has log discrepancy ≥ 0 with respect to (W, Ws" ), thus Ws has normal crossings in codimension one. In particular, the pair (W, Ws" ) is log canonical. Applying ‘Adjunction’ [Kaw07] and criterion A.1.3, we conclude that Ws" has semilog canonical singularities. The variety Ws" is the desired stable limit of Xη ; it is unique by the uniqueness of the canonical models. Now we construct our desired stack-theoretic stable limit. The point is that the stack-theoretic canonical model has canonical singularities for the same reason W has. In classical terms, since W has canonical singularities, its index-one cover does as well [KM98, 5.20, 5.21]. Furthermore, the direct-image sheaves [n]

ψ∗ ωψ ,

n ≥ 0,

are locally free and commute with restriction to the ﬁbers, i.e., [n]

[n]

ψ∗ ωψ |s" = Γ(Wt , ωWs" ). This formulation of ‘deformation invariance of plurigenera’ follows from the existence of minimal models [Nak86, Cor. 3, Thm. 8] and the good behavior of direct images of dualizing sheaves [Kol86, Thm. 2.1]. Since W is a canonical model, we can express [n]

W = ProjΔ" ⊕n≥0 ψ∗ ωψ . The corresponding aﬃne cone [n]

CW := SpecΔ" ⊕n≥0 ψ∗ ωψ

comes with a Gm -action associated with the grading, and we deﬁne W = [ (CW 1 0) / Gm ] . Again, we have Wη" @ Xη" (as polarized orbispaces) and thus Ws" has Gorenstein semilog canonical singularities. Indeed, W has canonical singularities (it is locally modeled by the index-one cover of W ) and thus is Cohen-Macaulay; the dualizing sheaf ωW/Δ" is invertible by the Gm -quotient construction. Repeating the previous adjunction analysis yields that Ws" is semilog canonical. The uniqueness of the stack-theoretic limit follows from the uniqueness of the canonical model, via Theorem 5.3.6. We turn now to boundedness: [Kar00, §3] still shows that each connected ω component of the closure of Kcan is of ﬁnite type: Matsusaka’s big theorem shows ω the open locus Kcan is of ﬁnite type. We apply weak semistable reduction to an arbitrary compactiﬁcation of a tautological family, and obtain a family X → B which is weakly semistable. In particular, B is smooth. Applying ﬁnite generation ([BCHM10]) we can take the relative canonical model X can = ProjB R(X/B) → B, or its orbispace version X can = Proj B R(X/B) → B. The total space has

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canonical singularities as it is a canonical model - this is true for either X can or X can . Deformation invariance of plurigenera shows that this is ﬂat. An argument similar to the one above shows that the ﬁbers are semilog canonical. This gives a compactiﬁed family over a base of ﬁnite type as required. The argument there, citing Koll´ ar’s projectivity criterion [Kol90], shows that ω properness of the closure of Kcan implies projectivity of its coarse moduli space. We would like the following formulation of ‘semi-canonical models’: Assumption 6.1.3. Suppose that Δ is the spectrum of a discrete valuation ring with special point s and generic point η. Let Xη be a stable variety over η. Then there exists a ﬁnite ramiﬁed base-change ΔD → Δ with special point sD and generic point η D , a unique variety W , and a ﬂat morphism ψ : W → ΔD satisfying the following: • Wη" is isomorphic to the base-change Xη" = X ×η η D . • The relative dualizing sheaf ωψ is Q-Cartier and ample. [n]

• Each reﬂexive power ωψ is S2 relative to ψ, i.e., S2 on restriction to the [n]

ﬁbers. If Xη is Cohen-Macaulay, we expect the sheaves ωψ to be CohenMacaulay as well [KK09]. • Ws" has semilog canonical singularities. [n]

• The sheaves ψ∗ ωψ , n ≥ 0, are locally free and satisfy [n]

[n]

ψ∗ ωψ |s" = Γ(Ws" , ωWs" ). ω Proposition 6.1.4. Under Assumption 6.1.3, Kslc satisﬁes the valuative criterion for properness. In particular, each connected component of ﬁnite type is proper. ω Remark 6.1.5. To say that each connected component of Kslc is of ﬁnite type is a boundedness assertion about the stable varieties with given numerical invariants. These assertions have been proven in dimensions ≤ 2 [Ale94], but remain open in higher dimensions.

Remark 6.1.6. The discussion following Corollary 1.3 of [KK09] implies that the stable limit of a one-parameter family of Cohen-Macaulay stable varieties (resp. Gorenstein twisted stable varieties) is Cohen-Macaulay (resp. Gorenstein). Thus ω ω Kgor-slc ⊂ Kslc is a union of connected components. Nevertheless, there are good reasons to consider non-Cohen-Macaulay singularities. We have already mentioned non-Cohen-Macaulay log canonical singularities in Remark 5.3.9. In characteristic zero, log terminal singularities and their indexone covers are automatically Cohen-Macaulay [KM98].

33

Stable varieties

A. The semilog canonical locus is open A.1. Statement. The following result is known for surfaces ([KSB88, Kol90]). The present proof was suggested to us by V. Alexeev. The second part can be found in [Kar00]. Proposition A.1.1. Let B be a scheme of ﬁnite type over a ﬁeld of characteristic zero and f : X → B be a ﬂat morphism of ﬁnite type with connected reduced equidimensional ﬁbers. Assume that the ﬁbers are S2 and Gorenstein in codimension one. Suppose there exists an invertible sheaf L on X such that for each b ∈ B we have ⊗n L|Xb @ j∗ ωU , n ∈ N, where j : U n→ Xb is the open subset where Xb is Gorenstein. Then the locus where the ﬁbers have semilog canonical singularities B slc = {b ∈ B : Xb is semilog canonical} is open in B. Remark A.1.2. In the case where the ﬁbers are Gorenstein, the existence of L is automatic. Indeed, the relative dualizing sheaf ωX/B is itself invertible. Before starting the proof proper, we recall a characterization of semilog canonical singularities, which can be found in chapter 2 of [Kol92]: Lemma A.1.3. Let (Y, D) be a pair consisting of a connected equidimensional S2 scheme Y and a reduced eﬀective divisor D with no components contained in the singular locus of Y . Suppose that Y has normal crossings in codimension one and that the Weil divisor KY +D is Q-Cartier. Let ν : Y˜ → Y denote the normalization and C ⊂ Y˜ its conductor, i.e., the reduced eﬀective divisor such that ˜ + C, ν ∗ (KY + D) = KY˜ + D ˜ is the proper transform of D. Then the following conditions are equivalent: where D • (Y, D) is semilog canonical. ˜ + C) is log canonical. • (Y˜ , D The condition of having normal crossings in codimension 1 is also open. Indeed, this means that the only singularities in codimension 1 are nodes, which can only deform to smooth points. We therefore assume that the ﬁbers of f have normal crossings in codimension one. A.2. Constructibility. Lemma A.2.1. Under the assumptions of Proposition A.1.1, the locus Bslc is constructible in B.

34

D. Abramovich and B. Hassett

Proof. The main ingredient is resolution of singularities. The image of a constructible set is constructible, so we may replace B by a resolution of singularities of its normalization B D → B. We have the pull back family X D = X ×B B D → B D . We claim we can stratify B D by locally-closed subsets & BjD BD = j∈J

such that the restriction over each stratum fj : XjD = X D |BjD → BjD admits a simultaneous good resolution of singularities. By deﬁnition, a good resolution of singularities is one where the preimage of the singular locus is a simple normal crossings divisor; in a simultaneous good resolution, each intersection of components of the simple normal crossings divisor is smooth over the base. The stratiﬁcation is constructed inductively: Choose a good resolution of singularities W → X D , which induces a good resolution of the geometric generic ﬁber of X D → B D . Since we are working over a ﬁeld of characteristic zero, such ﬁber is smooth. We can choose a dense open subset B DD ⊂ B D such that W → X D is a simultaneous good resolution of the ﬁbers XbD for b ∈ B DD . However, B D 1 B DD 2 B D is closed, so we are done by Noetherian induction. The ﬁbers of fj have invertible dualizing sheaves, so Lemma A.1.3 allows us to determine whether the ﬁbers have SLC singularities by computing discrepancies on the good resolutions. However, discrepancies are constant over families with a simultaneous good resolution, so if one ﬁber of fj is SLC then all ﬁbers are SLC. In particular, the ﬁbers are SLC over a constructible subset of the base. A.3. Proof of Proposition. We complete the proof that Bslc is open by proving that it is stable under one-parameter generalizations. Precisely, let T be a nonsingular connected curve, t ∈ T a closed point, µ : T → B a morphism, and fT : XT = X ×B T → T the base change of our family to T . Then the set {s ∈ T : Xs is semilog canonical } is open. The existence of the sheaf L guarantees that the canonical class of XT is Q-Cartier. Suppose that s ∈ T is such that Xs has semilog canonical singularities. Let ˜ T → XT denote the normalization and g : X ˜ → T the induced morphism. ν :X Let C denote the conductor divisor, which is also ﬂat over T . We may write ν ∗ KX = KX˜ + C,

35

Stable varieties

which is also Q-Cartier. If S is the normalization of some irreducible component ˜ s and ι : S → X ˜ s the induced morphism, then of X ι∗ (KX˜ s + Cs ) = KS + Θ, where Θ contains the conductor and the preimage of Cs . Lemma A.1.3 implies ˜ s , Cs ) is semilog canonical. By that (S, Θ) is log canonical, and therefore that (X ˜ X ˜ s + C) is log canonical as well, ‘Inversion of Adjunction’ [Kaw07], the pair (X, ˜ s . Fix a good log resolution at least in a neighborhood of X ˜ X ˜ s + C), ρ : (Y, D) → (X, i.e., one where the union of the exceptional locus and the proper transform of the boundary is simple normal crossings. All discrepancies are computed with respect ˜ s is a ﬁber of X ˜ → T , we have to this resolution. Since X ˜ C) for exceptional divisors with center in X ˜ s are • the log discrepancies of (X, ≥ 0; ˜ C) for exceptional divisors dominating T are • the log discrepancies of (X, ≥ −1; ˜ t is normal. Furthermore, ρ For t K= s in a neighborhood of s, the ﬁber X ˜ ˜ X ˜ t + C), which induces a good log resolution of (Xt , Ct ). Consider the pairs (X, are also log canonical by the discrepancy analysis above. Applying ‘Adjunction’, ˜ t , Ct ) is also log canonical. Lemma A.1.3 implies that Xt is we conclude that (X therefore semilog canonical.

References [Abr02]

Dan Abramovich, Canonical models and stable reduction for pluriﬁbered varieties. Unpublished, arXiv:math/0207004, 2002.

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J´ anos Koll´ ar and Sandor J. Kov´ acs, Log canonical singularities are Du Bois, 2009. Preprint arXiv:0902.0648.

[KKMSD73] G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I. Lecture Notes in Mathematics, Vol. 339. Springer-Verlag, Berlin, 1973. [KM97]

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J´ anos Koll´ ar, Flatness criteria. J. Algebra 175 (1995), 715–727.

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J. Koll´ ar and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities. Invent. Math. 91 (1988), 299–338.

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G´erard Laumon and Laurent Moret-Bailly, Champs alg´ebriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2000.

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Department of Mathematics, Box 1917, Brown University, Providence, RI, 02912, U.S.A E-mail: [email protected] Department of Mathematics, MS 136, Rice University, Houston, TX 77251, U.S.A E-mail: [email protected]

Basic properties of log canonical centers Florin Ambro∗

Abstract. We present the elementary properties of log canonical centers of log varieties. 2010 Mathematics Subject Classiﬁcation. Primary 14B05; Secondary 14E30. Keywords. Log variety, log canonical center.

1. Introduction Log varieties and their log canonical centers provide a natural setting for inductive arguments % in higher dimensional algebraic geometry. The prototype log variety is (X, i bi Ei ), where X is a nonsingular variety, the Ei ’s are nonsingular % prime divisors intersecting transversely, and bi ∈ [0, 1]. If bi < 1 for all i, (X, i bi Ei ) has so called Kawamata log terminal singularities, and it has no log canonical centers. In general, some of the Ei ’s will have coeﬃcient one, and the connected components of their intersections are called log canonical centers. Further, let C be a log canonical center which is a connected component of ∩i∈J Ei , where bi = 1 for all i ∈ J. Then a successive application of the classical adjunction formula gives the log adjunction formula M M (KX + bi Ei )|C = KC + bi Ei |C . i

i∈J /

Creating log canonical centers, restricting to them by adjunction, and lifting sections via vanishing theorems – these are the three steps of a powerful technique for constructing sections of adjoint line bundles in characteristic zero, parallel to the L2 -methods for singular hermitian metrics in complex geometry (see [11, 9, 15, 3] and [18, 7] for the algebraic and analytic side of the story, respectively). In this note we present the elementary properties of log canonical centers. They are easy to see in our example above: its log canonical centers are nonsingular, ﬁnite in number, their intersections are unions of log canonical centers, and their unions have seminormal singularities. Most of these properties extend to the general case of log varieties with log canonical singularities, by Shokurov [17], Koll´ ar [14], EinLazarsfeld [8], Kawamata [12] and [2], under some mild extra hypotheses, and [4] in general. Since they seem to be obscured by the new terminology of quasi-log varieties in [4], we reproduce them here. Theorem 1.1. Let (X, B) be a log variety with log canonical singularities, deﬁned over an algebraically closed ﬁeld of characteristic zero. Then: ∗ Research

supported by PCE Grant PNII 2228 (502/2009).

40

Florin Ambro

(1) (X, B) has at most ﬁnitely many log canonical centers. (2) An intersection of two log canonical centers is a union of log canonical centers. (3) Any union of log canonical centers has seminormal singularities. (4) Let x ∈ LCS(X, B) be a closed point. Then there is a unique minimal log canonical center Cx passing through x, and Cx is normal at x. In previous approaches, when there exists 0 ≤ B 0 ≤ B such that (X, B 0 ) has Kawamata log terminal singularities, Theorem 1.1 follows from KawamataViehweg’s vanishing. The new idea is to replace vanishing by the log canonical version of Koll´ar’s torsion freeness (which in turn follows from Esnault-Viehweg’s injectivity theorem).

2. Log varieties A log variety (X, B) is a normal variety X endowed with an eﬀective R-Weil divisor B such that KX +B is R-Cartier. We assume that X is deﬁned over an algebraically closed ﬁeld k, of characteristic zero. The canonical divisor KX is deﬁned as the Weil divisor (ω) of zeros and poles on X of a non-zero top rational diﬀerential form ω ∈ ∧dim X Ω1X ⊗k k(X) (it depends on the choice of ω, but only up to linear equivalence). Now B is a ﬁnite combination of prime divisors with real non-negative coeﬃcients, and %the R-Cartier hypothesis means that locally on X, KX + B equals a ﬁnite sum i ri (ϕi ), where ri ∈ R and ϕi ∈ k(X)× are non-zero rational functions on X. Let µ : X D → X be a birational morphism. Let ω be a non-zero top rational form on X. Then µ∗ ω is a non-zero top rational form on X D , and we can deﬁne an R-Weil divisor on X D by the formula BX " = µ∗ ((ω) + B) − (µ∗ ω). As any two non-zero top rational forms diﬀer by a rational function, it follows that BX " is independent of the choice of ω. If we denote KX = (ω) and KX " = (µ∗ ω), the deﬁnition of BX " becomes the so called log pull-back formula µ∗ (KX + B) = KX " + BX " . For a prime divisor E ⊂ X D , the real number 1 − multE (BX " ) is called the log discrepancy of (X, B) at E, denoted a(E; X, B). This number depends only on the µ valuation of k(X) deﬁned by E ⊂ X D → X. We call such a valuation geometric, and denote cX (E) = µ(E). Deﬁnition 2.1. The log variety (X, B) is said to have • log canonical singularities if a(E; X, B) ≥ 0 for every geometric valuation E of X.

41

Log canonical centers

• Kawamata log terminal singularities if a(E; X, B) > 0 for every geometric valuation E of X. The loci where (X, B) has log canonical and Kawamata log terminal singularities, respectively, are non-empty open subsets of X. We denote their complements by (X, B)−∞ and LCS(X, B), respectively. In particular, (X, B)−∞ ⊆ LCS(X, B), (X, B)−∞ = ∅ if and only if (X, B) has log canonical singularities, and LCS(X, B) = ∅ if and only if (X, B) has Kawamata log terminal singularities. Remark 2.2. Suppose that (X, B) does not have log canonical singularities. Then for every cycle C ⊆ (X, B)−∞ and every positive integer n, there exists a geometric valuation E of X such that a(E; X, B) < −n and cX (E) = C. This property motivates our choice of notation for the locus where (X, B) does not have log canonical singularities. Deﬁnition 2.3. A cycle C ⊂ X is called a log canonical center if (X, B) has log canonical singularities at the generic point of C, and there exists a geometric valuation E of X such that a(E; X, B) = 0 and cX (E) = C. %l Example 2.4. Let X be a nonsingular variety and B = i=1 bi Ei , where bi ∈ R≥0 and {Ei }li=1 are nonsingular prime divisors intersecting transversely. Let I0 = {i; bi = 1} and I−∞ = {i; bi > 1}. Then • The locus where (X, B) has Kawamata log terminal singularities is X \ ∪i∈I0 ∪I−∞ Ei . Its complement LCS(X, B) has a closed subscheme structure, with deﬁning ideal M OX (− \bi LEi ). i∈I0 ∪I−∞

• The locus where (X, B) has log canonical singularities is the open set X \ ∪i∈I−∞ Ei . • The log canonical centers of (X, B) are the connected components of the intersections ∩i∈J Ei , for ∅ = K J ⊆ I0 . Remark 2.5. Our notion of log canonical center diﬀers from the standard one used in the literature. The latter is deﬁned as a center C = cX (E), where E is a geometric valuation with a(E; X, B) ≤ 0. In our case, we further require that (X, B) has log canonical singularities at the generic point of C. The two notions coincide for log varieties with log canonical singularities, but diﬀer otherwise. For example, consider the log variety (C2 , H1 + 2H2 ), where Hi : (xi = 0). In the standard literature, H1 , H2 , and every point of H2 is a log canonical center. In our sense, only H1 is a log canonical center. This seems reasonable, given that KC2 + H1 + 2H2 cannot be restricted by adjunction to H2 , or any its points. This also shows that the ﬁniteness in Theorem 1.1.(1) fails in the non-log canonical case. Remark 2.6. By Hironaka, we may choose µ so that X D is non-singular, and l the proper transform µ−1 ∗ B and the µ-exceptional locus ∪i=1 Ei is supported by a

42

Florin Ambro

simple normal crossings divisor. We have the following formula BX " = µ−1 ∗ B+

l M

(1 − a(Ei ; X, B))Ei .

i=1

One can see that (X, B)−∞ is the image in X of the components of BX " with coeﬃcients in (1, +∞), and LCS(X, B) is the image in X of the components of BX " with coeﬃcients in [1, +∞). The lc centers of (X, B) are the sets µ(S), where S is a connected component of an intersection of components of BX " with multiplicity one, such that µ(S) 5 (X, B)−∞ . Remark 2.7. Canonical singularities were introduced by Reid (see [16]) as the singularities that appear on canonical models of projective manifolds of general type. Likewise, log canonical singularities are the singularities that appear on log canonical models of prototype log varieties of general type. It would be interesting to similarly describe semi-log canonical singularities (see [1]) and quasi-log canonical singularities (see [4]). To see this in the log canonical case, let (X, B) be a log variety with log canonical singularities. In the setting of Remark 2.6, the following formula holds: KX " + µ−1 ∗ B+

l M

Ei = µ∗ (KX + B) +

i=1

l M

a(Ei ; X, B)Ei .

i=1

%l By log canonicity, i=1 a(Ei ; X, B)Ei is eﬀective and µ-exceptional, so this formula becomes a relative Zariski decomposition for the µ-big log canonical divisor on the left-hand side, and we obtain . m∈N

µ∗ OX " (m(KX " + µ−1 ∗ B+

l M i=1

Ei )) =

.

OX (m(KX + B)).

m∈N

Further, if B is rational, (X, B) is recovered as follows: the graded OX -algebra on the left hand side is ﬁnitely generated (since the right-hand side is), its Proj %l is X, and B is the push forward of µ−1 ∗ B + i=1 Ei through the natural map X D ''( Proj. More generally, consider a log variety (X D , B D ) with log canonical singularities, D and a proper morphism π : X D → S such that KX " + >B is π-big and rational. D We expect that the graded OS -algebra R(X /S, B) = m∈N π∗ OX (m(KX " + B D )) is ﬁnitely generated. If so, we obtain a natural birational map Φ : X D ''( X := Proj(R(X D /S, B D )), deﬁned over S, and then (X, Φ∗ B D ) has log canonical singularities.

3. A torsion freeness theorem The aim of this section is Theorem 3.4, the log canonical version of Koll´ar’s torsion freeness [13]. It was proved in a more general setting in [4, Theorem 3.2.(i)], but

43

Log canonical centers

we reproduce its proof in the special case needed. Recall that the characteristic is zero. Theorem 3.4 is a direct consequence of the injectivity theorem of EsnaultViehweg [10, Theorem 5.1]. Unlike similar results of Tankeev or Koll´ ar, their result has the striking feature that it requires no positivity. Therefore we thought the reader might be interested to see the idea behind the proof. We reproduce it as Theorem 3.2, with Proposition 3.1 as the key technical statement behind it. We stress that Proposition 3.1 and Theorem 3.2 look diﬀerent from the corresponding statements in [10], but they are exactly the same (except the trivial extension from ∼Q - to ∼R -equivalence in Theorem 3.2.(i)). We just translated their statements into the language of torsion Q-divisors, a simple exercise proposed in [10, Remark 3.4.e)]. Their statements become more transparent, since they tie up naturally with log structures. The reader may check the equivalence by using the formulas in [10, Remark 3.4]. Proposition 3.1 ([10], Properties 2.9, Lemma 2.10, Theorem 3.2). Let X be a nonsingular projective variety and T a Q-divisor such that T ∼Q 0 and T − \T L = %l Let d1 , . . . , dl ∈ Z≥0 , and R a i=1 δi Ei has simple normal crossings support. % %l l reduced divisor with no common components with i=1 Ei , such that R + i=1 Ei has simple normal crossings. Denote E = OX (−R + \T L). Then the natural inclusion of complexes (with diﬀerentials induced by T ) Ω•X (log R +

l M

Ei ) ⊗ E(−

i=1

l M

di Ei ) → Ω•X (log R +

i=1

l M

Ei ) ⊗ E

i=1

is a quasi-isomorphism, and the spectral sequence E1pq = H q (X, ΩpX (log R +

l M i=1

Ei ) ⊗ E) =⇒ Hp+q (X, Ω•X (log R +

l M

Ei ) ⊗ E)

i=1

degenerates at E1 . Idea of proof. First, we see how T induces the diﬀerentials. Let n be a minimal positive integer with nT ∼ 0. There exists a rational function ϕ ∈ k(X)× such that (ϕ) = nT . Let π : X D → X be the normalization of X in the ﬁeld √ k(X)( n ϕ). There exists an open subset U ⊆ X, with complement of codimension at least two, % such that U D % = π −1 (U ) is nonsingular and the restriction to D D ∗ U of the support i" % Ei" of π ( i Ei ) is also a simple normal crossings divisor. Let d : OU " → Ω1U " (log i" EiD" |U " ) be the K¨ahler diﬀerential. The homomorphism π∗ (d) is compatible with action of the Galois group Z/nZ = mζC, so its component of eigenvalue ζ, denoted ∇, is an integrable connection on OX (\T L) with logarith% mic poles along i Ei . It is a priori deﬁned only on U , but it extends to X since OX (\T L) is locally free. Its residues are ResEi (∇) = δi · id : OX (\T L) ⊗ OEi → OX (\T L) ⊗ OEi . Now ∇ induces % an integral connection on OX (−R + \T L) with logarithmic poles along R + i Ei . Since the residue of this connection along each Ei is given by

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Florin Ambro

multiplication with the fractional number δj ∈ (0, 1), it follows from [10, Properties 2.9, Lemma 2.10] that the natural map of complexes is a quasi-isomorphism. The last statement follows from the degeneration of the spectral sequence associated to the Hodge ﬁltration on a logarithmic de Rham complex (Deligne [5, Corollary 3.2.13]), applied to some desingularization of X D . Theorem 3.2 ([10], Theorem 5.1). Let L, D be Cartier divisors on a nonsingular projective variety X. Assume that there are nonsingular divisors Ei intersecting transversely, and bi ∈ [0, 1] such that: % (i) L ∼R KX + i bi Ei . % (ii) D is eﬀective, supported by 0
H q (X, E(−D)) (

β

Hq (X, Ω•X (log

% i

Ei ) ⊗ E(−D))

α

! Hq (X, Ω• (log X

% i

Ei ) ⊗ E).

The ﬁrst part of Proposition 3.1 gives that α is an isomorphism, and the second part implies that β is surjective. We infer that H q (X, E(−D)) → H q (X, E) is surjective. −1 By Serre duality, H 0 (X, ωX ⊗E −1 ) → H 0 (X, ωX ⊗E% (D)) is injective. This is the −1 desired injective map, since ωX ⊗ E = OX (iL − 0 0. (iii) Supp(D) and Supp(DD ) do not contain log canonical centers of (X,

%

i bi Ei ).

45

Log canonical centers

Then the natural maps H q (X, OX (L)) → H q (X, OX (L+D)) are injective for all q. D Proof. By Hironaka, there exists a birational modiﬁcation that % µ : X → X such ∗ " ) − K , µ D and X D is projective and nonsingular, and BX " = µ∗ (KX + i bi E i X % µ∗ DD are all supported by a simple normal crossings divisor i" EiD" . Decompose BX " = B D − A into the positive and negative part. By assumption, µ∗ D and µ∗ (DD ) do not contain components of \B D L. Therefore ∗ ∗ D D there exists 0 < r > 1 such that \B D + %iAAD− A + rµ D + rµ (D )L = \B L. Since ∗ µ H is semiample, we may enlarge i" Ei" and assume that there % exists an Rdivisor H D with the following properties: H D is supported by i" EiD" , it has no common components with BX " , µ∗ D and µ∗ (DD ), \H D L = 0, H D ∼Q (1 − r)µ∗ H. We obtain

iAA + µ∗ L ∼Q KX " + B D + iAA − A + rµ∗ D + rµ∗ (DD ) + H D . Since the eﬀective Cartier divisor µ∗ D is supported by the fractional part of the boundary B D + iAA − A + rµ∗ D + rµ∗ (DD ) + H D , Theorem 3.2 gives the injectivity of the map H q (X D , OX " (iAA + µ∗ L)) → H q (X D , OX " (iAA + µ∗ L + µ∗ D)). On the other hand, µ∗ OX " (iAA) = OX and Rq µ∗ OX " (iAA) = 0 for q > 0. In particular, µ∗ OX " (iAA + µ∗ L) = OX (L) and Rq µ∗ OX " (iAA + µ∗ L) = 0 for q > 0. Since the Leray spectral sequence E2pq = H p (X, Rq µ∗ OX " (iAA + µ∗ L)) =⇒ H p+q (X D , OX " (iAA + µ∗ L)) ∼

degenerates, we obtain an isomorphism H q (X, OX (L))→H q (X D , OX " (iAA + µ∗ L)). ∼ We obtain an isomorphism H q (X, OX (L + D))→H q (X D , OX " (iAA + µ∗ L + µ∗ D)) in a similar way. The result follows. % Theorem 3.4. Let (X, i bi Ei ) be a log variety such that X is nonsingular, Ei are nonsingular divisors intersecting transversely, and bi ∈ [0, 1] for all i. Let f%: X → S be a proper morphism, and L a Cartier divisor on X such that L−KX − i bi Ei is f -semiample. Let q ≥ 0 and s a local section of Rq f∗ OX (L), which is zero%at the generic points of f (X) and f (C), for every log canonical center C of (X, i bi Ei ). Then s = 0. % Proof. Step 1. We may suppose L ∼R KX + i bi Ei . Indeed, the conclusion is local on S, % so near a ﬁxed ﬁber we may suppose that the f -semiample R-divisor % % L − KX − i bi Ei is in fact semiample. Then L − KX − i bi Ei = j rj H j , with rj ≥ 0 and |Hj | free linear systems.%Choose integers n > r and general j j % r members Dj ∈ |nj Hj |, and denote B = i bi Ei + j njj Dj . Then Supp B has % simple normal crossings, B has coeﬃcients in [0, 1], (X, i bi Ei ) and (X, B) have the same log canonical centers, % and L ∼R KX + B. Step 2. Let L ∼R KX + i bi Ei . We may suppose S is aﬃne and s ∈ Γ(S, Rq f∗ OX (L)). Then there exists a non-zero divisor h ∈ Γ(U, OS ) such h · s = 0

46

Florin Ambro

and D(h) contains the generic points of f (X) and f (C). Let A = (h) be the eﬀective Cartier divisor on S associated to h. Then s belongs to the kernel of the homomorphism Γ(S, Rq f∗ OX (L)) → Γ(S, Rq f∗ OX (L) ⊗ OS (A)). Suppose by contradiction that s K= 0. Then s/h K= 0 lies in the kernel of the homomorphism Γ(S, Rq f∗ OX (L) ⊗ OS (A)) → Γ(S, Rq f∗ OX (L) ⊗ OS (2A)). Then we may compactify X and S, and suppose A extends to a hyperplane section of S, still denoted by A, with the property that f ∗ A contains no log canonical % center of (X, i bi Ei ), and the following homomorphism is not injective Γ(S, Rq f∗ OX (L + f ∗ A)) → Γ(S, Rq f∗ OX (L + 2f ∗ A)). Consider the commutative diagram of spectral sequences E2p,q = H p (S, Rq f∗ OX (L + f ∗ A))

=⇒

H p+q (X, OX (L + f ∗ A))

# ¯ p,q = H p (S, OX (L + 2f ∗ A)) E 2

=⇒

# H p+q (X, OX (L + 2f ∗ A)).

The map E20,q → H q (X, OX (L + f ∗ A)) is injective, so we conclude that the homomorphism H q (X, OX (L + f ∗ A)) → H q (X, OX (L + 2f ∗ A)) is not injective. This contradicts Corollary 3.3.

4. Proof of Theorem 1.1 % Remark 4.1. Let X be a nonsingular variety, i∈I Ei a simple normal crossings divisor, I D , I DD non-empty subsets of I, and C a connected component of ∩i∈I " Ei . Then C ⊆ ∪i∈I "" Ei if and only if I D ∩ I DD K= ∅. Lemma 4.2. Let (X, B) be a log variety with log canonical singularities, and W a union of log canonical centers of (X, B). Let µ : X D → X be a resolution of singularities with log pullback µ∗ (K + B) = KX " + BX " , such that µ−1 (W ) is a divisor and µ−1 (W ) ∪ Supp(BX " ) has simple normal crossings. Let S be the union of prime divisors E on X D with multE (BX " ) = 1 and E ⊂ µ−1 (W ). Then OW = µ∗ OS . Proof. We have a unique decomposition BX " = S + R + Δ − A, where S consists of the components of BX " with coeﬃcient equal to one and which are included in µ−1 (W ), R consists of the components of BX " with coeﬃcient equal to one but which are not included in µ−1 (W ), Δ is the part of BX " with coeﬃcients in (0, 1)

Log canonical centers

47

and −A is the negative part of BX " . By Remark 4.1 applied to R and µ−1 (W ), no connected component of an intersection of components of R is mapped inside W . Consider the exact sequence µ∗ OX " (iAA) → µ∗ OS (iA|S A) → R1 µ∗ OX " (iAA − S). Because W = µ(S), iAA − S = KX " + R + iAA − A + Δ − µ∗ (K + B) and by Theorem 3.4, the last map is zero. Therefore µ∗ OX " (iAA) → µ∗ OS (iAA|S ) is surjective. Since B is eﬀective, we deduce that A is µ-exceptional. Therefore OX = µ∗ OX " (iAA), which implies µ∗ OS = OW . Proof of Theorem (1.1). (1) Choose µ : X D → X such that X D is smooth and BX " has simple normal crossings support. The log canonical centers are the images on X of the components of BX " with coeﬃcient one, and their intersections. Therefore they are ﬁnite. (2) Let C1 , C2 be two log canonical centers. By (1), it suﬃces to show that for every closed point x ∈ C1 ∩ C2 , there exists a new log canonical center x ∈ C3 ⊂ C1 ∪ C2 . Let W = C1 ∪ C2 . We may choose µ so that the hypotheses of Lemma 4.2 hold. In the notations of Lemma 4.2, we have µ∗ OS = OW . In particular, S → C1 ∪ C2 has connected ﬁbers. Therefore there are prime components E1 , E2 of S such that µ(Ei ) = Ci and E1 ∩ E2 ∩ π −1 (x) K= ∅. Let Z be a connected component of E1 ∩ E2 which intersects π −1 (x). Then C3 = µ(Z) is a log canonical center with x ∈ C3 ⊂ C1 ∪ C2 . (3) Let W be the union of some log canonical centers. We may choose µ so that the hypotheses of Lemma 4.2 hold. In the notations of Lemma 4.2, we have µ∗ OS = OW . Since S clearly has seminormal singularities, we infer by [2, Proposition 4.5] that W has seminormal singularities. (4) Fix x ∈ LCS(X, B) and consider (X, B) as a germ near x. By (1) and (3), there exists a unique log canonical center x ∈ C which is minimal with respect to inclusion. It remains to check that C is normal near x. Construct S as above with µ∗ OS = OC . Since C is minimal, every connected component of an intersection of components of S dominates C. Consider the simplicial scheme (Sn = (S0 /S)Δn → S)n≥0 , where S0 → S is the normalization (see [6]). Each irreducible component of Sn is nonsingular and mapped onto an intersection of components of S, hence it dominates C. Then each Sn → C factors through the normalization of C. These factorizations glue (cf. [4, Lemma 2.2.(ii)]), so that S → C factors through the normalization of C. But µ∗ OS = OC , so C is normal.

References [1] Alexeev, V., Limits of stable pairs. Preprint math.AG/0607684. [2] Ambro, F., The locus of log canonical singularities. Preprint math.AG/9806067. [3] Ambro, F., The Adjunction Conjecture and its applications. PhD Thesis (February 1999), The Johns Hopkins University; math.AG/9903060.

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[4] Ambro, F., Quasi-log varieties. In Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220–239; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 214–233. ´ [5] Deligne, P., Th´eorie de Hodge. II. Inst. Hautes Etudes Sci. Publ. Math. 40 (1971), 5–57. ´ [6] Deligne, P., Th´eorie de Hodge. III, Inst. Hautes Etudes Sci. Publ. Math. 44 (1974), 5–77. [7] Demailly, J.-P., Multiplier ideal sheaves and analytic methods in algebraic geometry. In School on Vanishing Theorems and Eﬀective Results in Algebraic Geometry (Trieste, 2000), 1–148, ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. [8] Ein, L.; Lazarsfeld, R., Global generation of pluricanonical and adjoint linear series on smooth projective threefolds. J. Amer. Math. Soc. 6 (1993), 875–903. [9] Ein, L., Multiplier ideals, vanishing theorems and applications. In Algebraic geometry—Santa Cruz 1995, 203–219, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997. [10] Esnault, H.; Viehweg, E., Lectures on vanishing theorems. DMV Seminar, 20. Birkh¨ auser Verlag, Basel, 1992. [11] Kawamata, Y.; Matsuda, K.; Matsuki, K., Introduction to the minimal model program, In Algebraic Geometry, Sendai, Advanced Studies in Pure Math. 10 (1987), 283–360. [12] Kawamata, Y., On Fujita’s freeness conjecture for 3-folds and 4-folds. Math. Ann. 308 (1997), 491–505. [13] Koll´ ar, J., Higher direct images of dualizing sheaves. I. Ann. of Math. 123 (1986), 11–42. [14] Koll´ ar, J., Adjunction and discrepancies. In Flips and abundance for algebraic threefolds. Ast´erisque No. 211 (1992). [15] Koll´ ar, J., Singularities of pairs. In Algebraic geometry—Santa Cruz 1995, 221–287, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997. [16] Reid, M., Young person’s guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345–414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. [17] Shokurov, V. V., Three-dimensional log perestroikas. With an appendix in English by Yujiro Kawamata. Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203; translation in Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. [18] Siu, Y.-T., The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. In Geometric complex analysis (Hayama, 1995), 577–592, World Sci. Publ., River Edge, NJ, 1996. Florin Ambro, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. BOX 1-764, RO-014700 Bucharest, Romania E-mail: ﬂ[email protected]

Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces I. Bauer, F. Catanese∗

Abstract. This is the ﬁrst in a series of articles on the moduli spaces of Burniat surfaces. We prove that Burniat surfaces are Inoue surfaces and we calculate their fundamental groups. As a main result of this paper we prove that every smooth projective surface which is homotopically equivalent to a primary Burniat surface (i.e., a Burniat surface with K 2 = 6) is indeed a primary Burniat surface. 2010 Mathematics Subject Classiﬁcation. 14J29, 14J25, 14J10, 14D22, 14H30, 20F34, 32G05, 32Q30 Keywords. Moduli of surfaces of general type, fundamental groups of algebraic varieties.

Introduction In a recent joint paper ([BCGP09]) with Fritz Grunewald and Roberto Pignatelli we constructed many new families of surfaces of general type with pg = 0, hence we got interested about the current status of the classiﬁcation of such surfaces, in particular about the structure of their moduli spaces. For instance, in the course of deciding which families were new and which were not new, we ran into the problem of determining whether surfaces with K 2 = 4 and with fundamental group equal to the one of Keum-Naie surfaces were indeed KeumNaie surfaces. This problem was solved in [BC09a], where we showed that any surface homotopically equivalent to a Keum-Naie surface is a Keum-Naie surface, whence we got a complete description of a connected irreducible component of the moduli space of surfaces of general type. We soon realized that similar methods would apply to the ’primary’ Burniat surfaces, the ones with K 2 = 6; hence we got interested about the components of the moduli space containing the Burniat surfaces. This article is the ﬁrst of a series of articles devoted to the so called Burniat surfaces. These are several families of surfaces of general type with pg = 0, K 2 = 6, 5, 4, 3, 2, ﬁrst constructed by P. Burniat in [Bu66] as ’bidouble covers’ (i.e., (Z/2Z)2 Galois covers) of the plane P2 branched on certain conﬁgurations of nine lines. These surfaces were later considered by Peters in [Pet77], who gave an account of Burniat’s construction in the modern language of double covers. He missed ∗ The

present work took place in the realm of the DFG Forschergruppe 790 ”Classiﬁcation of algebraic surfaces and compact complex manifolds”.

50

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however one of the two families with K 2 = 4, the ’non nodal’ one. He also calculated (ibidem) the torsion group H1 (S, Z) for Burniat’s surfaces (observe that a surface of general type with pg = 0 has ﬁrst Betti number b1 = 0). He asserted 2 that H1 (S, Z) ∼ = (Z/2Z)KS . This result is however correct only for K 2 K= 2, as we shall see. Later, following a suggestion by Miles Reid, another construction of these surfaces was given by Inoue in [In94], who constructed ’surfaces closely related to Burniat’s surfaces’ with a diﬀerent technique as G2 := (Z/2Z)3 -quotients of a ˆ of multidegree (2, 2, 2) in a product of three elliptic G2 -invariant hypersurface X curves. Another description of the Burniat surfaces as ’singular bidouble covers’ was later given in [Cat99], where also other examples were proposed of ’Burniat type surfaces’. These however turn out to give no new examples. The important feature of the Burniat surfaces S is that their bicanonical map is a bidouble cover of a normal Del Pezzo surface of degree KS2 (obtained as the anticanonical model of the blow up of the plane in the points of multiplicity at least 3 of the divisor given by the union of the lines of the conﬁguration). Burniat surfaces with K 2 = 6 were studied from this point of view by MendesLopes and Pardini in [MLP01]. Although Burniat surfaces had been known for a long time, we found that their most important properties were yet to be discovered, and we devote two articles to show in particular that the four families of Burniat surfaces, the ones with K 2 = 6, 5 respectively, and the two ones with K 2 = 4 (the nodal and the non nodal one) are irreducible connected components of the moduli space of surfaces of general type. Since there is no reference known where it is proved that Burniat’s surfaces are exactly Inoue’s surfaces, we start by giving in the present paper a proof of this fact. This is crucial in order to calculate the fundamental groups of Burniat’s surfaces with K 2 = 6, 5, 4, 3, 2. Our proof conﬁrms the results stated by Inoue without proof in his beautiful paper, except for K 2 = 2 where Inoue’s claim turns out to be wrong. Our proof combines the ’transcendental’ description given by Inoue with delicate algebraic calculations, which are based on explicit algebraic normal forms for the 2-torsion of elliptic curves, described in the ﬁrst section. We ﬁrst prove the following: Theorem 0.1. Let S be the minimal model of a Burniat surface. i) KS2 = 6 =⇒ π1 (S) = Γ, H1 (S, Z) = (Z/2Z)6 ; 2 2 ii) 3 ≤ KS2 ≤ 5 =⇒ π1 (S) = H ⊕ (Z/2Z)K −2 , H1 (S, Z) = (Z/2Z)K ; iii) K 2 = 2 =⇒ π1 (S) = H1 (S, Z) = (Z/2Z)3 . Here H denotes the quaternion group of order 8, while Γ is a group of aﬃne transformations on C3 , explicitly described in section 3. The main result of this article is however the following theorem:

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Burniat surfaces I

Theorem 0.2. Let S be a smooth complex projective surface which is homotopically equivalent to a primary Burniat surface. Then S is a Burniat surface. We can then use this result to give an alternative, and less involved proof of the following result due to Mendes-Lopes and Pardini ([MLP01]). Theorem 0.3. The subset of the Gieseker moduli space corresponding to primary Burniat surfaces is an irreducible connected component, normal, rational and of dimension equal to 4. In [MLP01] openness is shown (using Burniat’s description of a primary Burniat surface) by standard local deformation theory of bidouble covers. We give an alternative proof of this result using Inoue’s description. For the closedness, Mendes Lopes and Pardini use their characterization of primary Burniat surfaces as exactly those surfaces with pg = 0, K 2 = 6 such that the bicanonical map has degree 4. Our proof is much less involved. It only uses the description of the fundamental group as an aﬃne group of transformations of C3 . Moreover we prove the rationality of the moduli space of primary Burniat surfaces. In the second article we shall show that the Burniat surfaces with K 2 = 5 yield an irreducible connected component of dimension 3 in the moduli space of surfaces of general type. Instead, there are two diﬀerent conﬁgurations in the plane giving Burniat surfaces with K 2 = 4. We shall show that, in fact, both yield an irreducible connected component of dimension 2 in the moduli space of surfaces of general type. This is interesting, since it follows that the bicanonical map of S is a bidouble cover of a Del Pezzo surface of degree KS2 for all the surfaces in the connected component. There is only one Burniat surface with K 2 = 2, and since its fundamental group is (Z/2Z)3 , it turns out to be a surface in the 6 dimensional family of standard Campedelli surfaces ([Miy77]), for which the bicanonical map is then a degree 8 covering of the plane. We analyse it brieﬂy in the last section. In fact, at the moment of completing the paper we became aware of the article [Ku04] where the author had already pointed out and corrected the errors of [In94] and [Pet77] on the fundamental group and the ﬁrst homology of the Burniat surface with K 2 = 2.

1. The Legendre and other normal forms for 2-torsion of elliptic curves This section reviews classical mathematics which will be reiteratedly used in the sequel. The Legendre form of an elliptic curve is given by an equation of the form y 2 = (ξ 2 − 1)(ξ 2 − a2 ).

52

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It yields a curve E D of genus 1 as a double cover of P1 branched on the 4 points ξ = ±1, ξ = ±a. D D These 4 points yield 4 points on E D , P1D , P−1 , PaD , P−a , which correspond to the 2-torsion points, once any of them is ﬁxed as the origin, as we shall more amply now illustrate. We consider now 3 automorphisms of order 2 of E D deﬁned by: g1D (ξ, y) := (−ξ, −y),

g2D (ξ, y) := (ξ, −y),

g3D (ξ, y) := (−ξ, y).

We get in this way an action of (Z/2Z)2 on E D such that the quotient is P1 , with coordinate u := ξ 2 . Clearly the quotient by g2D is the original P1 with coordinate ξ, hence g2D corresponds to multiplication by −1 on the elliptic curve, once we ﬁx one of the above points as the origin. The quotient of E D by g3D is instead the smooth curve of genus 0, given by the conic y 2 = (u − 1)(u − a2 ). What is more interesting is the quotient of E D by g1D : the invariants are u and r := ξy, thus we obtain as quotient the elliptic curve E := {r2 = u(u − 1)(u − a2 )} in Weierstrass normal form. This shows that g1D is the translation by a 2-torsion element η D . By looking at the action of g1D on the 4 above points, we see that η D is the class of the degree zero D divisor [P1D ] − [P−1 ]. In other words, the divisor classes of degree 0 D D ], ] = [PaD ] − [P−a η D := [P1D ] − [P−1

D D ] ] − [P−a η DD := [P1D ] − [PaD ] = [P−1

generate P ic0 (E D )[2] ∼ = (Z/2Z)2 . We can now also understand that the automorphism g3D , which is the product D D g1 g2 = g2D g1D , has as ﬁxed points the 4 points lying over ξ = 0 and ξ = ∞. These correspond to the 4-torsion points whose associated translation has as square the translation by the 2-torsion point η D . More importantly, we want to give now a nicer form for the group of translations of order 2 of an elliptic curve (this leads to the theory of 2-descent on an elliptic curve). This form will be used in the sequel, but in another coordinate system for the line P1 with coordinate ξ. To this purpose, let us consider the curve C deﬁned by v 2 = (ξ 2 − 1), w2 = (ξ 2 − a2 ). We shall show that this curve is the same elliptic curve E which we had above. In fact, setting y := vw, we see that we obtain C as a double cover of E D , which is unramiﬁed (as we see by calculating the ramiﬁcation of the (Z/2Z)2 -Galois cover of P1 with coordinate ξ). The transformations of order 2 g1 : (ξ, v, w) ?→ (ξ, −v, −w), f2 : (ξ, v, w) ?→ (−ξ, v, −w),

53

Burniat surfaces I

f3 : (ξ, v, w) ?→ (−ξ, −v, w) ∼ generate a group H = (Z/2Z)2 such that the quotient curve is the elliptic curve E, since the invariants are ξ 2 = u, v 2 = u − 1, w2 = u − a2 , ξvw = ξy = r. The quotients by the above involutions are respectively E D , the elliptic curve E DD := {t2 = (v 2 + 1)(v 2 + 1 − a2 )} (where we have set t := ξw), and the elliptic curve E DDD := {s2 = (w2 + a2 )(w2 + a2 − 1)} (where we have set s := ξv). The ﬁrst conclusion is that C is isomorphic to E, the group H is the group of translations by the 2-torsion points of E, whereas the quotient map C → E = C/H is multiplication by 2 in the elliptic curve C ∼ = E. We have another group G ∼ = (Z/2Z)2 acting on C ∼ = E, namely the one with quotient the P1 with coordinate ξ. Here, we set g2 : (ξ, v, w) ?→ (ξ, −v, w), g3 := g1 g2 = g2 g1 : (ξ, v, w) ?→ (ξ, v, −w). Again, g1 corresponds to translation by a 2-torsion element η, while we view g2 as multiplication by −1. The ﬁxed points √ of g2 are the points with v = 0, i.e., the 4 points with v = 0, ξ = ±1, w = ± 1 − a2 . Translation by η then acts on them simply by multiplying their w coordinate by −1. An important observation is that the covering C → P1 , where P1 has coordinate u, is a (Z/2Z)3 -Galois cover of the P1 with coordinate u which is the maximal Galois covering of P1 branched on the 4 points 0, 1, a2 , ∞ and with group of the form (Z/2Z)m . It would be nice if also for surfaces one could treat such Galois covers with group (Z/2Z)m in the same elementary way . This however can be done only in the birational setting, since in dimension ≥ 2 we have diﬀerent normal models for the same function ﬁeld. Hence we have to resort to the theory of abelian covers, developed in [Cat84], [Par91], [Cat99]. In this biregular theory, coverings are described through equations holding in certain vector bundles. To compare the surface case with the curve case it is therefore useful ﬁrst of all to rewrite the above (Z/2Z)3 -Galois cover in terms of homogeneous coordinates. And, for later calculations, it will be convenient to replace the points ξ = ±1 with the points 0, ∞. We replace then the aﬃne coordinates (1 : ξ) by coordinates (xD : x) with xx" = ξ−1 ξ+1 . We can then rewrite the (Z/2Z)2 cover as the normalization of the curve in 1 P × P1 × P1 given by 1 2 2 2 2 {(v D : v), (wD : w), (xD : x) | v 2 xD = v D x, w2 xD = wD (x2 − xxD (b + ) + xD )}. b Now the involution exchanging pairs of branch points is simply the involution (xD : x) → (x : xD ). The normalization is obtained simply by considering the curve

54

I. Bauer, F. Catanese

of genus 1 which is the subvariety of the vector bundle whose sheaf of sections on P1 is OP1 (1) ⊕ OP1 (1), given by equations 1 2 V 2 = xxD , W 2 = (x2 − xxD (b + ) + xD ), b which is shorthand notation for the following two equations in the local chart outside xD = 0, respectively in the local chart outside x = 0: ( (

x w 2 x 2 1 x v 2 ) = ( D ), ( D ) = ( D ) − (b + ) D + 1, D v x w x b x 2

2

2

vD xD w xD xD 1 xD ) = ( ), ( D ) = ( ) − (b + ) + 1. v x w x x b x 2

In other words, we have v 2 = x, v D = xD , hence V = vv D . While, setting W := 2 ( ww" )xD , we get W 2 = (x2 − xxD (b + 1b ) + xD ). We have now the group (Z/2Z)3 acting on C by the following transformations g1 : ((xD : x), (v D : v), (wD : w)) ?→ ((xD : x), (v D : −v), (wD : −w)), f2 : ((xD : x), (v D : v), (wD : w)) ?→ ((x : xD ), (v : v D ), (wD x : −wxD )), f3 : ((xD : x), (v D : v), (wD : w)) ?→ ((x : xD ), (−v : v D ), (wD x : wxD )), g2 : ((xD : x), (v D : v), (wD : w)) ?→ ((xD : x), (v D : −v), (wD : w)), g3 = g1 g2 : ((xD : x), (v D : v), (wD : w)) ?→ ((xD : x), (v D : v), (wD : −w)). The sections V and W are clearly eigenvectors for the group action. It is easy to see, in view of the above table, that the image of V = vv D equals −V, V, −V, −V, V respectively, while the image of W equals −W, −W, W, W, −W respectively.

2. Burniat surfaces are Inoue surfaces The aim of this section is to show that Burniat surfaces are Inoue surfaces. This fact seems to be known to the experts, but, since we did not ﬁnd any reference, we shall provide a proof of this assertion, which is indeed crucial for our main result. In [Bu66], P. Burniat constructed a series of families of surfaces of general type with K 2 = 6, 5, 4, 3, 2 and pg = 0 (of respective dimensions 4, 3, 2, 1, 0) as singular bidouble covers (Galois covers with group (Z/2Z)2 ) of the projective plane branched on 9 lines. We brieﬂy recall the construction. ˆ 2 (P1 , P2 , P3 ) Let P1 , P2 , P3 ∈ P2 be three non collinear points and denote by Y := P 2 the blow up of P in P1 , P2 , P3 . Y is a Del Pezzo surface of degree 6 and it is the closure of the graph of the rational map r : P2 ''( P1 × P1 × P1 such that

r(y1 : y2 : y3 ) = ((y2 : y3 ), (y3 : y1 ), (y1 : y2 )).

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Burniat surfaces I

It is immediate to observe that Y ⊂ P1 × P1 × P1 is the following hypersurface of type (1, 1, 1): Y = {((xD1 : x1 ), (xD2 : x2 ), (xD3 : x3 )) | x1 x2 x3 = xD1 xD2 xD3 }. Lemma 2.1. Consider the cartesian diagram p−1 (Y ) # P1 × P1 × P1

p

!Y i

p

# ! P1 × P1 × P1

where p : P1 × P1 × P1 → P1 × P1 × P1 is the (Z/2Z)3 -Galois covering given by xi = vi2 , xDi = (viD )2 . Then p−1 (Y ) splits as the union p−1 (Y ) = Z ∪ Z D of two degree 6 Del Pezzo surfaces, where Z := {((v1 : v1D ), (v2 : v2D ), (v3 : v3D )) : v1 v2 v3 = v1D v2D v3D } and

Z D := {((v1 : v1D ), (v2 : v2D ), (v3 : v3D )) : v1 v2 v3 = −v1D v2D v3D }.

And p|Z induces on P2 the Fermat squaring map (y0 : y1 : y2 ) ?→ (y02 : y12 : y22 ). Moreover, Z ∩Z D = {v1 v2 v3 = v1D v2D v3D = 0}, which is the union of 6 lines yielding in each Del Pezzo surface the fundamental hexagon of the blow up of P2 in three non collinear points (i.e., the pull back of the triangle with vertices the three points). Proof. The equation of p−1 (Y ) is x1 x2 x3 = xD1 xD2 xD3 , i.e., (v1 v2 v3 )2 = (v1D v2D v3D )2 . The surface Z is invariant under the subgroup Go ⊂ { 1, −1}3 ∼ = (Z/2Z)3 , Go = {(ri ) ∈ {±1}3 |

Go ∼ = (Z/2Z)2 , 1

ri = 1}.

i

Go acts on Z by sending vi → ? ri vi , viD → ? viD , and this is easily seen to give on P2 the Galois group of the Fermat squaring map. We denote by Ei the exceptional curve lying over Pi and by Di,1 the unique eﬀective divisor in |L − Ei − Ei+1 |, i.e., the proper transform of the line yi−1 = 0, side of the triangle joining the points Pi , Pi+1 . For the present choice of coordinates Ei is the side xi−1 = xDi+1 = 0 of the hexagon, while Di,1 is the side xi = xDi+1 = 0 of the hexagon.

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Consider on Y the following divisors Di = Di,1 + Di,2 + Di,3 + Ei+2 ∈ |3L − 3Ei − Ei+1 + Ei+2 |, where Di,j ∈ |L − Ei |, for j = 2, 3, Di,j = K Di,1 , is the proper transform of another line through Pi and Di,1 ∈ |L − Ei − Ei+1 | is as above. % Assume also that all the corresponding lines in P2 are distinct, so that D := i Di is a reduced divisor. Observe that all the indices in {1, 2, 3} have here to be understood as residue classes modulo 3. Note that, if we deﬁne the divisor Li := 3L − 2Ei−1 − Ei+1 , then Di−1 + Di+1 = 6L − 4Ei−1 − 2Ei+1 ≡ 2Li , and we can % consider (cf. [Cat99]) the associated bidouble cover X → Y branched on D := i Di (but with diﬀerent ordering of the indices: we take here one which is more apt for our notation). We recall that this precisely means the following: let Di = div(δi ), and let ui be a ﬁbre coordinate of the geometric line bundle Li , whose sheaf of holomorphic sections is OY (Li ). Then X ⊂ L1 ⊕ L2 ⊕ L3 is given by the equations: u1 u2 = δ1 u3 , u21 = δ3 δ1 ; u2 u3 = δ2 u1 , u22 = δ1 δ2 ; u3 u1 = δ3 u2 , u23 = δ2 δ3 . From the birational point of * view, we are * simply adjoining to the function ﬁeld Δ1 2 2 of P two square roots, namely Δ3 and Δ Δ3 , where Δi is the cubic polynomial in C[x0 , x1 , x2 ] whose zero set has Di as strict transform. This shows clearly that we have a Galois cover with group (Z/2Z)2 . The equations above give a biregular model X which is nonsingular exactly if the divisor D does not have points of multiplicity 3 (there cannot be points of higher multiplicities). These points give then quotient singularities of type 14 (1, 1), i.e., the quotient of C2 by the action of (Z/4Z) sending (u, v) → ? (iu, iv) (or, equivalently , the aﬃne cone over the 4-th Veronese embedding of P1 ). This (cf. [Cat08] for more details) can be seen by an elementary calculation. Assume in fact that δ1 , δ2 , δ3 are given in local holomorphic coordinates by x, y, x− y, and that we deﬁne locally wi as the square root of δi . Then: w12 = x, w22 = y, w32 = x − y ⇒ w32 = w12 − w22 . Therefore the singularity is an A1 singularity, quotient of C2 by the action of (Z/2Z) sending (u, v) → ? (−u, −v) (here, w3 = uv, u2 = w1 + w2 , v 2 = w1 − w2 ). The action of (Z/4Z) on C2 induces the action of (Z/2Z) on the A1 singularity, ? −wi , ∀i. Finally, the functions ui = wi+1 wi+2 and wi2 = δi generate sending wi → the (Z/4Z)-invariants, subject to the linear relation δ1 − δ2 = δ3 . The singularity can be resolved by blowing up the point x = y = 0, and then the inverse image of the exceptional line is a smooth rational curve with self intersection −4.

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Deﬁnition 2.2. A primary Burniat surface is a surface constructed as above, and which is moreover smooth. It is then a minimal surface S with KS ample, and with KS2 = 6, pg (S) = q(S) = 0. A secondary Burniat surface is a surface constructed as above, and which moreover has 1 ≤ m ≤ 2 singular points (necessarily of the type described above). Its minimal resolution is then a minimal surface S with KS nef and big, and with KS2 = 6 − m, pg (S) = q(S) = 0. A tertiary Burniat surface is a surface constructed as above, and which moreover has 3 ≤ m ≤ 4 singular points (necessarily of the type described above). Its minimal resolution is then a minimal surface S with KS nef and big, and with KS2 = 6 − m, pg (S) = q(S) = 0. Remark 2.1. 1) We remark that for KS2 = 4 there are two possible types of conﬁgurations. The one where there are three collinear points of multiplicity at least 3 for the plane curve formed by the 9 lines leads to a Burniat surface S which we call of nodal type, and with KS not ample, since the inverse image of the line joining the 3 collinear points is a (-2)-curve (a smooth rational curve of self intersection −2). In the other cases with KS2 = 4, 5, instead, KS is ample. 2) In the nodal case, if we blow up the two (1, 1, 1) points of D, we obtain a weak Del Pezzo surface, since it contains a (-2)-curve. Its anticanonical model has a node (an A1 -singularity, corresponding to the contraction of the (-2)-curve). In the non nodal case, we obtain a smooth Del Pezzo of degree 4. This fact has obviously been overlooked by [Pet77], since he only mentions the nodal case. In the sequel to this paper we shall show that in the case of secondary Burniat surfaces with KS2 = 4 these two families indeed give two diﬀerent connected components of dimension 2 in the moduli space. And also that secondary Burniat surfaces with KS2 = 5 form a connected component of dimension 3 in the moduli space. 3) We illustrate the possible conﬁgurations in the plane in ﬁgure 1. In [In94] Inoue constructed a series of families of surfaces with K 2 = 6, 5, 4, 3, 2 and pg = 0 (of respective dimensions 4, 3, 2, 1, 0, exactly as for the Burniat surfaces) as the (Z/2Z)3 quotient of an invariant hypersurface of type (2, 2, 2) in a product of three elliptic curves. As already mentioned, it seems to be known to the specialists that these Inoue’s surfaces are exactly the Burniat’s surfaces, but for lack of a reference we show here: Theorem 2.3. Burniat’s surfaces are exactly Inoue’s surfaces. Proof. Consider as in lemma 2.1 the cartesian diagram p−1 (Y ) # P1 × P1 × P1

p

!Y i

p

# ! P1 × P1 × P1

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I. Bauer, F. Catanese

K2=5

P3

K2=6

P3

P4

P1

P2

K2=4 nodal

P1

P2

K2=4 nonnodal

P3

P3

P4 P5

P5

P4 P1

P2

K2=3

P2

P1

K2=2 P3

P4 P6

P4

P7

P5

P5

P3 P6 P1

P1

P2

Figure 1. Conﬁgurations of lines

P2

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Burniat surfaces I

where p : P1 × P1 × P1 → P1 × P1 × P1 is the (Z/2Z)3 -Galois covering given by xi = vi2 , xDi = (viD )2 . Then p−1 (Y ) splits as the union of two degree 6 Del Pezzo surfaces p−1 (Y ) = Z ∪ Z D , where Z := {((v1 : v1D ), (v2 : v2D ), (v3 : v3D )) : v1 v2 v3 = v1D v2D v3D } and

Z D := {((v1 : v1D ), (v2 : v2D ), (v3 : v3D )) : v1 v2 v3 = −v1D v2D v3D }.

F Recall that the subgroup of (Z/2Z)3 stabilizing Z is Go = {(ri ) ∈ {±1}3 | i ri = 1}. We can further extend the previous diagram by considering a (Z/2Z)6 Galoiscovering pˆ : E1 × E2 × E3 → P1 × P1 × P1 obtained by taking, with diﬀerent choices of the (xD : x) coordinates, the direct product of three (Z/2Z)2 Galois-coverings Ei → P1 as in section 1. What we have now explained is summarized in the bottom two lines of the ˆ is deﬁned as the inverse image of the following commutative diagram, where X Del Pezzo surface Z. ˆ X

G2

! X = X/G ˆ 2 $$$ $$$ $$$ $$$ $$) ! Z ∪ ZD !Y

ˆi

# ˆ ∪X ˆD X # E1 × E2 × E3

(Z/2)3

# ! P1 × P1 × P1

(Z/2)3

# ! P1 × P1 × P1 .

ˆ ∪X ˆ D n→ E1 × E2 × E3 is the inclusion of X ˆ ∪X ˆ D as Note that the vertical map i : X a divisor of multidegree (4, 4, 4) splitting as a union of two divisors of respective multidegrees (2, 2, 2). ˆ is a (Z/2Z)3 Galois covering of X, ramiﬁed only Next we want to show that X 1 in the points of type 4 (1, 1) (and hence ´etale in the case of a primary Burniat X). ˆ is In fact, the stabilizer of X 1 ri = 1} ∼ G1 := {(ri , rDi ) ∈ {±1}3 × {±1}3 | = (Z/2Z)5 . i

ˆ a (Z/2Z)5 Galois covering of Y , and we claim that we The action of G1 makes X obtain X as an intermediate cover by setting ui = Wi−1 Wi vi viD . Let us denote (Di,2 +Di,3 ) by DiD . This is the divisor deﬁned by a section δiD = 0 which is the pull back of a homogeneous polynomial of degree 2 on the ith copy of 2 P1 (this polynomial is the polynomial (x2i − xi xDi (bi + b1i ) + xDi ) in the notation of section 1). Let us then write Di = (Di,1 + Ei+2 ) + (Di,2 + Di,3 ) = Di,1 + Ei+2 + DiD .

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Observe that div(xi ) = Di,1 + Ei+1 , div(xDi ) = Di−1,1 + Ei−1 , whence D D Di + Di−1 = DiD + Di−1 + Di,1 + Ei+2 + Di−1,1 + Ei+1 = div(δiD δi−1 xi xDi ).

Now, the (Z/2Z)2 Galois- covering of the ith copy of P1 is given by: (viD vi )2 = xi xDi , Wi2 = δiD . Since we see that

u2i = δi δi−1 , D xi xDi = (Wi Wi−1 vi viD )2 . u2i = δi δi−1 = δiD δi−1

Whence we have established our claim that setting ui = Wi−1 Wi vi viD , ˆ → X. we get a mapping X ˆ → X is Galois with Galois group the subgroup G2 < G1 We also see that X leaving each ui invariant, which, by the above formulae, is given by {(ri , rDi ) ∈ G1 |rDi−1 rDi ri = 1, i ∈ {1, 2, 3}} ∼ = (Z/2Z)3 . The last isomorphism follows since the rDi ’s determine ri = rDi−1 rDi . A natural basis for G2 ≤ G1 ≤ (Z/2Z)6 ∼ = (Z/2Z)3 ⊕ (Z/2Z)3 is given by             1 1 0 0 0 1 (0 , 1) =: g1 , (1 , 1) =: g2 , (0 , 0) =: g3 . 0 0 0 1 1 1 Therefore, if zi is a uniformizing parameter for the elliptic curve Ei , with zi = 0 corresponding to the origin of Ei , we see that the action of G2 on E1 × E2 × E3 (cf. section 1) is given as follows:             −z1 z1 z1 z1 z1 + η1 z1 g1 z2  =  −z2  , g2 z2  = z2 + η2  , g2 z2  =  z2  . z3 z3 + η3 z3 −z3 z3 z3 ˆ → X is an ´etale (Z/2Z)3 Remark 2.2. If X is a primary Burniat surface, then X covering. Instead, for each (1, 1, 1) - point of D = D1 + D2 + D3 , X has a singular point ˆ → X is ramiﬁed in exactly these singular points, yielding 4 of type 14 (1, 1), and X ˆ nodes on X for each one of these singular points on X. ˆ is a divisor of type (2, 2, 2) invariant by the action of G2 , we have seen Since X that any Burniat surface X is an Inoue surface. ˆ 2 is an Inoue surface: since every such G2 Conversely, assume that X = X/G ˆ invariant surface X is the pull back of a Del Pezzo surface Zc := {(viD , vi )|v1 v2 v3 − cv1D v2D v3D = 0}, we see that X is a Burniat surface.

(1)

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Burniat surfaces I

Remark 2.3. In the above equation (1) there is a constant c appearing, whereas in the previous description we had normalized this constant to be equal to 1. On each P1 there are the points vi = 0, viD = 0, hence these coordinates are determined up to a constant λi . In turn, we have two more branch points, forming the locus of zeroes of an equation which we normalized as being vi2 + (bi + b1i ) vi viD + 2

viD = 0. This normalization now determines the constant λi uniquely, and ﬁnally F with these choice of coordinates we get the equation (1) with c = i λi , and we see that c is a function of b1 , b2 , b3 .

3. The fundamental groups of Burniat surfaces The aim of this section is to combine our and Inoue’s representation of Burniat surfaces in order to calculate the fundamental groups of the Burniat surfaces with K 2 = 6, 5, 4, 3, 2. In [In94] the author gave a table of the respective fundamental groups, but without supplying a proof. As we shall now see, his assertion is right for K 2 = 6, 5, 4, 3 but wrong for the case K 2 = 2. So we believe it worthwhile to give a detailed proof, especially in order to cast away any doubt on the validity of his assertion for K 2 = 6, 5, 4, 3. Let (E, o) be any elliptic curve, and consider as in section 1 the G = (Z/2Z)2 = {0, g1 , g2 , g3 := g1 g2 } - action given by g1 (z) := z + η, g2 (z) = −z, where η ∈ E is a 2 - torsion point of E. Remark 3.1. The divisor [o] + [η] ∈ Div 2 (E) is invariant under G, hence the invertible sheaf OE ([o] + [η]) carries a natural G-linearization. In particular, G acts on the vector space H 0 (E, OE ([o] + [η])) which splits then as a direct sum . H 0 (E, OE ([o] + [η]))χ H 0 (E, OE ([o] + [η])) = χ∈G∗

of the eigenspaces corresponding to the characters χ of G. We shall use a self explanatory notation: for instance, H 0 (E, OE ([o]+[η]))+− is the eigenspace corresponding to the character χ such that χ(g1 ) = 1, χ(g2 ) = −1. We recall the following: Lemma 3.1 ([BC09a], lemma 2.1). Let E be as above. Then H 0 (E, OE ([o] + [η])) = H 0 (E, OE ([o] + [η]))++ ⊕ H 0 (E, OE ([o] + [η]))−− . I.e., H 0 (E, OE ([o] + [η]))+− = H 0 (E, OE ([o] + [η]))−+ = 0.

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Let now Ei := C/Λi , i = 1, 2, 3, be three complex elliptic curves, and write Λi = Zei ⊕ ZeDi . Deﬁne now aﬃne transformations γ1 , γ2 , γ3 ∈ A(3, C) as follows:             z1 z1 z1 z1 −z1 z1 + e21 γ1 z2  =  −z2  , γ2 z2  = z2 + e22  , γ3 z2  =  z2  , z3 z3 z3 −z3 z3 z3 + e23 and let Γ ≤ A(3, C) be the aﬃne group generated by γ1 , γ2 , γ3 and by the translations by the vectors e1 , eD1 , e2 , eD2 , e3 , eD3 . Remark 3.2. Γ contains the lattice Λ1 ⊕ Λ2 ⊕ Λ3 , hence Γ acts on E1 × E2 × E3 inducing a faithful action of G2 := (Z/2Z)3 on E1 × E2 × E3 . We prove next the following Theorem 3.2. Let S be the minimal model of a Burniat surface. i) K 2 = 6 =⇒ π1 (S) = Γ, H1 (S, Z) = (Z/2Z)6 ; ii) K 2 = 5 =⇒ π1 (S) = H ⊕ (Z/2Z)3 , H1 (S, Z) = (Z/2Z)5 ; iii) K 2 = 4 =⇒ π1 (S) = H ⊕ (Z/2Z)2 , H1 (S, Z) = (Z/2Z)4 ; iv) K 2 = 3 =⇒ π1 (S) = H ⊕ (Z/2Z), H1 (S, Z) = (Z/2Z)3 ; v) K 2 = 2 =⇒ π1 (S) = H1 (S, Z) = (Z/2Z)3 . Here H denotes the quaternion group of order 8. Remark 3.3. As already said, these results conﬁrm, except for the case K 2 = 2, 2 the results of Inoue [In94], stating that for K 2 ≤ 5, π1 (S) = H ⊕ (Z/2Z)K −2 . Proof. i) Let S be the minimal model of a Burniat surface with KS2 = 6. Then, ˆ which by the previous section 2, S = X has an ´etale (Z/2Z)3 Galois covering X, is a hypersurface of multidegree (2, 2, 2) in the product of three elliptic curves ˆ is smooth and ample, by Lefschetz’s theorem π1 (X) ˆ = E1 × E2 × E3 . Since X π1 (E1 × E2 × E3 ) ∼ = Z6 . Γ acts on the universal covering of E1 × E2 × E3 ∼ = C3 , and acts freely on ˜ ⊂ C3 , the universal covering of X, ˆ with quotient the invariant hypersurface X ˜ ˜ is also the universal covering of S = X and π1 (S) = Γ. S = X = X/Γ. Hence X Next we shall prove that H1 (S, Z) = (Z/2Z)6 . Since γi2 = ei , for i = 1, 2, 3, it follows that Γ is generated by g1 , g2 , g3 , eD1 , eD2 , eD3 . It is clear that a) γ1 commutes with e1 , eD1 , e3 , eD3 ; b) γ2 commutes with e2 , eD2 , e1 , eD1 ; c) γ3 commutes with e2 , eD2 , e3 , eD3 .

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Burniat surfaces I

Writing tei ∈ A(3, C) for the translation by the vector ei , we see that −1 γ1 te2 = t−1 e2 γ1 , γ1 te"2 = te" γ1 ; 2

−1 γ2 te3 = t−1 e3 γ2 , γ2 te"3 = te" γ2 ; 3

γ3 te1 =

t−1 e1 γ3 ,

γ3 te"1 =

t−1 e"1 γ3 .

This implies that 2e1 , 2eD1 , 2e2 , 2eD2 , 2e3 , 2eD3 ∈ [Γ, Γ]. Moreover,     z1 z1 + e21 γ1 γ2 z2  = −z2 − e22  = t−1 e2 γ2 γ1 , z3 −z3 whence e2 ∈ [Γ, Γ]. Similarly, we see that (as the respective commutators of γ1 with γ3 , γ2 with γ3 ) e1 , e3 ∈ [Γ, Γ]. Therefore ΓD := Γ/me1 , e2 , e3 , 2eD1 , 2eD2 , 2eD3 C surjects onto Γab . But ΓD is already abelian, since the morphism ΓD → (Z/2Z)3 ⊕ (Z/2Z)3 , mapping the residue classes of γ1 , γ2 , γ3 , eD1 , eD2 , eD3 onto the ordered set of coordinate vectors of (Z/2Z)3 is easily seen to be well deﬁned and an isomorphism. This shows that H1 (S, Z) = (Z/2Z)6 . In order to prove the assertions ii) − v) observe preliminarly that, if X is the above singular model of S, then, by van Kampen’s theorem, π1 (S) ∼ = π1 (X). Therefore, for the remaining cases, it suﬃces to calculate π1 (X). Let X be the above singular model of a Burniat surface with K 2 ≤ 5. Consider ˆ Since the singularities of X ˆ are only nodes, the G2 ∼ = (Z/2Z)3 -Galois cover X. 6 ∼ ˆ π1 (X) = Z by the theorem of Brieskorn-Tyurina (cf. [Brie68], [Brie71], [Tju70]). By [Arm65], [Arm68] π1 (X) ∼ = Γ/ Tors(Γ), where Tors(Γ) is the normal subgroup of Γ generated by all elements of Γ having ﬁxed points on the universal covering ˜ of X ˆ (which is, as we have seen before, a Γ-invariant hypersurface in C3 ). X Note that the elements in G2 ∼ = (Z/2Z)3 induced by the elements γ1 , γ2 , γ3 , γ1 γ2 , γ1 γ3 , γ2 γ3 ∈ Γ do not have ﬁxed points on E1 × E2 × E3 . Instead,    −z1 + z1 γ1 γ2 γ3 z2  = −z2 − z3 −z3 −



e1 2 e2  2 e3 2

has as ﬁxed points the 64 points on E1 × E2 × E3 corresponding to vectors in C3 such that ei 2zi ≡ mod Λi , ∀i. (2) 2 Equivalently, ei 1 zi ≡ mod Λi , ∀i. (3) 4 2

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ˆ has 4 ii) Let X be the singular model of a Burniat surface with K 2 = 5. Then X 2 nodes (lying over the point P4 ∈ P , see ﬁgure 1. ˜ then there is a We observed that if γ ∈ Γ has a ﬁxed point on X, λ ∈ Z6 ∼ = me1 , e2 , e3 , eD1 , eD2 , eD3 C =: Λ such that γ = γ1 γ2 γ3 tλ . ˜ ⊂ C3 . Then z yields a ﬁxed point of γ1 γ2 γ3 on X ˆ if Let now z = (z1 , z2 , z3 ) ∈ X ˆ and only if there is a λ ∈ Λ such that     e1 z1 ˆ 1 ˆ ⇐⇒ z = 1 r + λ , 2 z2  = −e2  + λ 2 4 2 z3 −e3 

 e1 where we have set r := −e2 . −e3 ˆ In fact: We show now that z is a ﬁxed point of a γ as above iﬀ λ = −λ. 1 γ(z) = γ1 γ2 γ3 tλ (z) = −(z + λ) + r = 2 ˆ 1 λ 1 ˆ − r − − λ + r = z − λ − λ. 4 2 2 ˆ gets Modifying z modulo Λ, we replace z by z + λD , and the corresponding λ ˆ + 2λD ; hence we see that γ1 γ2 γ3 tλ has a ﬁxed point on X ˜ for all replaced by λ ˆ + 2Λ. Therefore 2Λ is contained in Tors(Γ). Since the above arguments λ ∈ −λ apply to all remaining cases (ii) - (v) we summarize what we have seen in the following ˆ (i.e., we are Lemma 3.3. If Γ has a ﬁxed point z on the universal covering of X ¯ in one of the cases ii) - v)), then π1 (X) is a quotient of Γ := Γ/2Λ. We have thus an exact sequence ¯ → (Z/2Z)3 → 1. 1 → (Z/2Z)6 → Γ In particular, we already showed that the fundamental group of a Burniat surface with K 2 ≤ 5 is ﬁnite: we are now going to write its structure explicitly. ¯ are contained in the Remark 3.4. 1) The images of ei , eDj , i, j ∈ {1, 2, 3} in Γ ¯ center of Γ, i.e., the above exact sequence yields a central extension. 2) Note that over each (1, 1, 1) point of the branch divisor D ⊂ P2 there are 4 nodes ˆ which are a G2 -orbit of ﬁxed points of γ1 γ2 γ3 on X. ˆ Let z ∈ X ˜ induce a of X, ˆz λ 1 ˆ ﬁxed point of γ1 γ2 γ3 on X: then z = 4 r + 2 , and the other 3 ﬁxed points in the ˆ for i = 1, 2, 3. We have: orbit are exactly the points induced by γi (z) on X,

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

γ1 (z) =

 e1 1 ˆ 4 + 2 (λz )1 1 ˆ 2  γ1  −e 4 − 2 (λz )2 −e3 1 ˆ 4 + 2 (λz )3

This implies that ˆ γ (z) λ 1

  e1 ˆ z + e2  ≡λ 0

=

ˆ γ (z) λ 1 r+ 1 . 4 2

mod 2Λ,

and similarly ˆ γ (z) λ 2

  0 ˆ z + e2  ≡λ e3

mod 2Λ,

ˆ γ (z) λ 3

  e1 ˆz +  0  ≡λ e3

mod 2Λ.

3) Let X be the singular model of a Burniat surface, and choose w.l.o.g. one of the ˆ z = 0. ˜ over the (1, 1, 1) - point P4 to be z := 1 r. This is equivalent to λ points in X 4 1 For each (1, 1, 1) point of the branch divisor D ⊂ P choose one singular point of ˆ lying over it. X Let S := {z(4) = z, . . . , z(9 − K 2 )} be a choice of representatives for each 2 ˆ lying over the respective (1, 1, 1) - points. Then: G -orbit of points of X ˆ z + 2Λ, z ∈ SC. π1 (X) = Γ/mγ1 γ2 γ3 tλ : λ ∈ −λ In particular, we have the relations: γ1 γ2 γ3 = 1, and, by 2) :

e1 = e2 = e3 .

Recall now that γi2 = ei . Therefore in π1 (X) we have: γ12 = γ22 = γ32 = e1 + e2 + e3 . Thus we get an exact sequence (cf. Lemma 3.3): 1 → (Z/2Z)3 ⊕ (Z/2Z) → π1 (X) → (Z/2Z)2 → 1, where the map ϕ : π1 (X) → (Z/2Z)2 is given by γ1 ?→ (1, 0), γ2 ?→ (0, 1), eDi ?→ 0. This immediately shows that the kernel of ϕ is equal to meD1 , eD2 eD3 , e1 +e2 +e3 = γi2 C. Let H := {±1, ±i, ±j, ±k} be the quaternion group, and let γ1 , γ2 , γ3 correspond respectively to i, j, −k: then we obtain an isomorphism π1 (X) ∼ = H ⊕ (Z/2Z)3 . This proves the assertion on the fundamental group for Burniat surfaces with K 2 = 5. That H1 (S, Z) = (Z/2Z)5 follows, since Hab = (Z/2Z)2 .

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iii), iv) First observe that, by the above, if X is the singular model of a Burniat surfaces with K 2 ≤ 5, then π1 (X) is the quotient of H ⊕ (Z/2Z)3 by the relations ˆ z(i) − λ ˆ z(j) = 0, where z(i) = λ K z(j) ∈ S. Note that for K 2 = 4 (nodal or non nodal), the projections of z(4) and z(5) to E2 , resp. E3 , are points whose diﬀerences are non trivial 2-torsion elements. Since each of the corresponding (1, 1, 1) points lies on two diﬀerent lines D2,2 , D2,3 , respectively D3,2 , D3,3 , the images of z(4) and z(5) under the composition of the projection to E2 (resp. to E3 ) with the quotient map E2 → P1 (resp. E3 → P1 ) have diﬀerent x2 -value (resp. x3 -value). ˆ z(4) − λ ˆ z(5) in ⊕3 eD Z/2Z is non zero. Claim. The image of λ i=1 i

Proof of the claim. Again, we look at the image of z(4) (resp. z(5)) in E2 → P1 (with coordinate of P1 equal to x2 ). We have seen that the corresponding (1, 1, 1) points P4 , P5 lie on two diﬀerent lines D2,2 , D2,3 , respectively D3,2 , D3,3 ; hence the respective x2 coordinates of the projection of z(4) and z(5) to P1 are diﬀerent. We conclude since the description of the transformations of order 2 of E2 given by translation by 2-torsion elements (cf. section 1) shows that translation by e22 is the only one which leaves the x2 coordinate invariant. QED for the claim. Therefore, if X is the singular model of a Burniat surface with K 2 = 4, π1 (X) is the quotient of H ⊕ (Z/2Z)3 by an element having a non trivial component in (Z/2Z)3 , hence π1 (X) ∼ = H ⊕ (Z/2Z)2 . Assume now that X is the singular model of a Burniat surface with K 2 = 3. Here the branch divisor on P2 has three (1, 1, 1)-points. Repeating the above ˆ z(4) − λ ˆ z(5) and argument, we see that π1 (X) is the quotient of H ⊕ (Z/2Z)3 by λ ˆ z(4) − λ ˆ z(6) . λ D As above, we look at the image in  ⊕3i=1 Z/2Z  ei  and see that they give (up to a 0 1 permutation of indices) the elements 1, 0. This implies that π1 (X) is the 1 1 quotient of H ⊕ (Z/2Z)3 by two linear independent relations in (Z/2Z)3 . Therefore π1 (X) = H ⊕ (Z/2Z). v) Let X be the singular model of a Burniat surfaces with K 2 = 2. Remark 3.5. Observe that, by [Rei], [Rei79], [Miy77], |π1 (X)| ≤ 9. Since π1 (X) is a quotient of H ⊕ (Z/2Z) by a relation coming from an element of Λ there are only two possibilities: either π1 (X) = H or π1 (X) = (Z/2Z)3 . We are going to show that the second alternative holds. Here the branch divisor D on P2 has four (1, 1, 1)-points P4 , P5 , P6 , P7 . As above, π1 (X) is the quotient of H ⊕ (Z/2Z)3 by the relations: ˆ z(4) − λ ˆ z(5) = 0, λ ˆ z(4) − λ ˆ z(6) = 0, λ ˆ z(4) − λ ˆ z(7) = 0, λ ˆ is a point lying over Pj . where z(j) ∈ X Looking at ﬁgure 1 of the conﬁguration of lines in P2 for the Burniat surface with K 2 = 2, we see that P4 , P5 ∈ D1,2 , P4 , P6 ∈ D2,2 , P4 , P7 ∈ D3,2 ,

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i.e., P4 , P5 lie in the same dashed line, but in two diﬀerent dotted and two diﬀerent black lines, P4 , P6 lie in the same dotted line, but in two diﬀerent dashed and two diﬀerent black lines, and P4 , P7 lie in the same black line, but in two diﬀerent dashed and two diﬀerent dotted lines. ˆ z(4) − λ ˆ z(5) , λ ˆ z(4) − λ ˆ z(6) , λ ˆ z(4) − λ ˆ z(7) This means that if we look at the image of λ 3 D in ⊕i=1 ei Z/2Z we see that they give (up to a permutation of indices) the elements       0 1 1 1 , 0 , 1 . 1 1 0 These three vectors are linearly dependent, hence, taking the quotient by these relations, the rank of ⊕3i=1 eDi Z/2Z drops only by two. ˆ z(4) − λ ˆ z(5) , λ ˆ z(4) − λ ˆ z(6) , In order to determine the component of the image of λ ˆ ˆ λz(4) − λz(7) in the center of the quaternion group, we have to write the points z(j), j ∈ {4, 5, 6, 7} more explicitly, using section 1. Observe that in the case K 2 = 2, we have E1 = E2 = E3 =: E. The ﬁxed √ points of γ1 γ2 γ3 are given by wi = 0, i = 1, 2, 3. Setting xDi = 1, and a := b, we can assume w.l.o.g. that   (1 : b), (1 : a), (1 : 0) z(4) = (1 : b), (1 : a), (1 : 0) . (1 : b), (1 : a), (1 : 0) By equation (1), we have v1 v2 v3 = cv1D v2D v3D , whence c = a3 . W.l.o.g., by (2) of remark 3.4, we can assume that z(5) is given by    (1 : b), (1 : ζ), (1 : 0) (1 : b), (1 : ζ), (1 : 0) z(5) = (b : 1), (a : 1), (1 : 0) = f2 ((1 : b), (1 : a), (1 : 0)) . (b : 1), (a : 1), (1 : 0) f2 ((1 : b), (1 : a), (1 : 0)) 

We have now to determine ζ in such a way that v1 v2 v3 = a3 v1D v2D v3D . For z(5) we have v1 v2 v3 = ζ = cv1D v2D v3D = a3 a2 = a5 .

(4)

Since by section 1 the only two translations of order 2 leaving (xD : x) unchanged are the identity and g1 , we have ((1 : b), (1 : ζ), (1 : 0)) = ((1 : b), (1 : a), (1 : 0)) or ((1 : b), (1 : ζ), (1 : 0)) = g1 ((1 : b), (1 : a), (1 : 0)) = ((1 : b), (1 : −a), (1 : 0)). Together with equation (4) we get: a5 = ζ = ±a, i.e., a4 = ±1. a4 = 1 is not possible, because this would imply that b = a2 = ±1, a contradiction to b K= 1b .

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Hence we have that a4 = −1, i.e., ζ = a5 = −a, and we see that the relation ˆ ˆ z(5) = 0 is given by λz(4) − λ   3 0 . eDi (Z/2Z) ⊕ Z/2Z, m1 , 1C ∈ i=1 1 where the last summand is the center of the quaternion group. Using for z(6) and z(7) the same argument as for z(5), we get two further elements which have to be set equal to zero in the quotient:     3 1 1 . m0 , 1C, m1 , 1C ∈ eDi (Z/2Z) ⊕ Z/2Z. i=1 1 0 >3 Taking the sum of these three elements in i=1 eDi (Z/2Z) ⊕ Z/2Z we see that we get   0 m0 , 1C = 0, 0 and we have concluded the proof of the theorem. Remark 3.6. We shall show in the last section how Burniat surfaces with K 2 = 2 are classical Campedelli surfaces, i.e., obtained as the tautological (Z/2Z)3 Galois covering of P2 branched on seven lines.

4. The moduli space of primary Burniat surfaces In this section we ﬁnally devote ourselves to the main result of the paper. First of all, we show Theorem 4.1. The subset of the Gieseker moduli space corresponding to primary Burniat surfaces is an irreducible connected component, normal, rational and of dimension equal to 4. This result was already proven in [MLP01] using the fact that the bicanonical map of the canonical model X D of a Burniat surface is exactly the bidouble covering 2 X D → Y D onto the normal Del Pezzo surface Y D of degree KX " obtained as the anticanonical model of the weak Del Pezzo surface obtained blowing up not only the points P1 , P2 , P3 , but also all the other triple points of D. We shall now give an alternative proof of their theorem. Proof. The singular model X of a primary Burniat surface is smooth, and has ample canonical divisor. Hence it equals the minimal model S (and the canonical model X D ).

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Burniat surfaces I

ˆ → X is ´etale with group G2 , it suﬃces to show that the Kuranishi Since X ˆ is smooth. Then it will also follow that the Kuranishi family of X family of X is smooth, whence the Gieseker moduli space is normal (being locally analytically isomorphic to the quotient of the base of the Kuranishi family by the ﬁnite group Aut(X)). ˆ ⊂ E1 × E2 × E3 is a smooth hypersurface, setting for convenience Since X T := E1 × E2 × E3 , we have the tangent bundle sequence 3 ˆ 0 → ΘXˆ → ΘT ⊗ OXˆ ∼ = OX ˆ (X) → 0 ˆ → OX

with exact cohomology sequence ˆ ∼ 0 → C3 → H 0 (OXˆ (X)) = C10 → ˆ ∼ → H 1 (ΘXˆ ) → H 1 (ΘT ⊗ OXˆ ) ∼ = C9 → H 1 (OXˆ (X)) = C3 . ˆ moves in a smooth family of dimension 13 = 6 + 7, a ﬁbre bundle Since X over the family of deformations of the principally polarized Abelian variety T , ˆ with ﬁbre the linear system P(H 0 (T, OT (X))), it suﬃces to show that the map 1 1 ˆ H (ΘT ⊗ OXˆ ) → H (OXˆ (X)) is surjective. ˆ ∼ It suﬃces to observe that H 1 (ΘT ⊗ OXˆ ) ∼ = H 2 (OT ), = H 1 (ΘT ), H 1 (OXˆ (X)) and that, as well known, the above map corresponds via these isomorphisms to the ˆ an element of H 1 (Ω1 ) which represents contraction with the ﬁrst Chern class of X, T a non degenerate alternating form. Whence surjectivity follows. ˆ is smooth (moreover the KodairaThus the base of the Kuranishi family of X Spencer map of the above family is a bijection, but we omit the veriﬁcation here), whence the base of the Kuranishi family of X, which is the G2 -invariant part of ˆ is also smooth. the base of the Kuranishi family of X, Moreover the Kuranishi family of X ﬁbres onto the family of G2 -invariant deformations of T , which coincides with the deformations of the three individual elliptic curves. The ﬁbres of the corresponding morphism between the bases of the respective ˆ which we are families are given by the G2 -invariant part of the linear system |X|, 1 going to calculate explicitly as being isomorphic to P . We obtain thereby a rational family of dimension 4 parametrizing the primary Burniat surfaces. This proves the unirationality of the 4 dimensional irreducible component. That the irreducible component of the moduli space is in fact a connected component follows from the more general result below (theorem 4.2). We calculate now H 0 (E1 × E2 × E3 , p∗1 OE1 ([o1 ] + [

2 e2 e3 e1 ]) ⊗ p∗2 OE2 ([o2 ] + [ ]) ⊗ p∗3 OE3 ([o3 ] + [ ]))G , 2 2 2

where pi : E1 × E2 × E3 → Ei is the i - th projection. From lemma 3.1 it follows that

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e1 ])) = H1+++ ⊕ H1−+− = 2 e1 e1 = H 0 (E1 , OE1 ([o1 ] + [ ]))+++ ⊕ H 0 (E1 , OE1 ([o1 ] + [ ]))−+− , 2 2

H1 := H 0 (E1 , OE1 ([o1 ] + [

e2 ])) = H2+++ ⊕ H2−−+ = 2 e2 e2 = H 0 (E2 , OE2 ([o2 ] + [ ]))+++ ⊕ H 0 (E2 , OE2 ([o2 ] + [ ]))−−+ , 2 2

H2 := H 0 (E2 , OE2 ([o2 ] + [

e3 ])) = H3+++ ⊕ H3+−− 2 e3 e3 = H 0 (E3 , OE3 ([o3 ] + [ ]))+++ ⊕ H 0 (E3 , OE3 ([o3 ] + [ ]))+−− . 2 2

H3 := H 0 (E3 , OE3 ([o3 ] + [

As a consequence of this, we get 2 e1 e2 e3 ]) ⊗ p∗2 OE2 ([o2 ] + [ ]) ⊗ p∗3 OE3 ([o3 ] + [ ]))G = 2 2 2 (H1+++ ⊗ H2+++ ⊗ H3+++ ) ⊕ (H1−+− ⊗ H2−−+ ⊗ H3+−− ) ∼ = C2 .

H 0 (E1 × E2 × E3 , p∗1 OE1 ([o1 ] + [

We have obtained a 4 - dimensional rational family parametrizing all the primary Burniat surfaces. This can also be seen in a more direct fashion by the fact that, ﬁxing 4 points in P2 in general position, we can ﬁx the 3 lines Di,1 , i = 1, 2, 3 and 2 lines D1,2 , D2,2 . Then the other 4 lines vary each in a pencil, hence we get 4 moduli. In the remaining part of this section, we will prove the following result: Theorem 4.2. Let S be a smooth complex projective surface which is homotopically equivalent to a primary Burniat surface. Then S is a Burniat surface. Proof. Let S be a smooth complex projective surface with π1 (S) = Γ. Recall that γi2 = ei for i = 1, 2, 3. Therefore Γ = mγ1 , eD1 , γ2 , eD2 , γ3 , eD3 C and we have the exact sequence 1 → Z6 ∼ = me1 , eD1 , e2 , eD2 , e3 , eD3 C → Γ → (Z/2Z)3 → 1, where ei → ? γi2 . If we set Λi := Zei ⊕ ZeDi , i = 1, 2, 3 then π1 (E1 × E2 × E3 ) = Λ1 ⊕ Λ2 ⊕ Λ3 . We also have the three lattices ΛDi := Z e2i ⊕ ZeDi .

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Remark 4.1. 1) Γ is a group of aﬃne transformations on ΛD1 ⊕ ΛD2 ⊕ ΛD3 . 2) We have an ´etale double cover Ei = C/Λi → EiD := C/ΛDi , which is the quotient by the semiperiod e2i of Ei . Γ has the following three subgroups of index two: Γ3 := mγ1 , eD1 , e2 , eD2 , γ3 , eD3 C, Γ1 := mγ1 , eD1 , γ2 , eD2 , e3 , eD3 C, Γ2 := me1 , eD1 , γ2 , eD2 , γ3 , eD3 C, corresponding to three ´etale double covers of S: Si → S, for i = 1, 2, 3. Lemma 4.3. The Albanese variety of Si is EiD . In particular, q(S1 ) = q(S2 ) = q(S3 ) = 1. Proof. Observe once more that i) γ1 commutes with e1 , eD1 , e3 , eD3 ; ii) γ2 commutes with e2 , eD2 , e1 , eD1 ; iii) γ3 commutes with e2 , eD2 , e3 , eD3 . Denoting by tei ∈ A(3, C) the translation with vector ei , we see that −1 γ1 te2 = t−1 e2 γ1 γ1 te"2 = te" γ1 ; 2

−1 γ3 te1 = t−1 e1 γ3 γ3 te"1 = te" γ3 . 1

This implies that 2e2 , 2eD2 , 2e1 , 2eD1 ∈ [Γ3 , Γ3 ]. Moreover, γ1 γ3 = t−1 e1 γ3 γ1 , whence already

surjective homomorphism

e1 ∈ [Γ3 , Γ3 ]. Therefore we have a

ΓD3 := Γ3 /m2e2 , 2eD2 , e1 , 2eD1 C = Γ3 /(2Z3 ⊕ Z) → Γab 3 = Γ3 /[Γ3 , Γ3 ]. Since the images of γ3 and eD3 are in the centre of ΓD3 , we get that ΓD3 is abelian, D hence H1 (S3 , Z) = Γab 3 = Γ3 and ΓD3 = mγ3 , eD3 C ⊕ (Z/2Z)4 = Z

e3 ⊕ ZeD3 ⊕ (Z/2Z)4 = Λ3 ⊕ (Z/2Z)4 . 2

This implies that Alb(S3 ) = C/ΛD3 = E3D . D 4 The same calculation shows that Γab i = H1 (Si , Z) = Λi ⊕ (Z/2Z) , whence D D Alb(Si ) = C/Λi = Ei , also for i = 2, 3. Let now Sˆ → S be the ´etale (Z/2Z)3 - covering associated to the subgroup Z ∼ = me1 , eD1 , e2 , eD2 , e3 , eD3 C i Γ. Since Sˆ → Si → S, and Si maps to EiD (via the Albanese map), we get a morphism 6

f : Sˆ → E1D × E2D × E3D = C/ΛD1 × C/ΛD2 × C/ΛD3 .

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Since the covering of E1D × E2D × E3D associated to Λ1 ⊕ Λ2 ⊕ Λ3 < ΛD1 ⊕ ΛD2 ⊕ ΛD3 is E1 × E2 × E3 , we see that f factors through E1 × E2 × E3 and the Albanese map of Sˆ is α ˆ : Sˆ → E1 × E2 × E3 . ˆ ⊂ T = E1 × E2 × E3 be the Albanese image of S. ˆ Let Y := α ˆ (S) We consider for i K= j ∈ {1, 2, 3}, the natural projections πij : E1 × E2 × E3 → Ei × Ej . Claim 4.4. If S is homotopically equivalent to a primary Burniat surface, then for i K= j ∈ {1, 2, 3} we have deg(πij ◦ α ˆ ) = 2. Moreover the cohomology class of the image Y equals 2F1 + 2F2 + 2F3 , where Fi is the pull back of a point in the elliptic curve Ei . Proof. The degree of πij ◦ α ˆ is the index of the image of H 4 (Ei × Ej , Z) inside 4 ˆ H (S, Z). But the former equals ∧4 (Λi ⊕ Λj ), hence we see that this number is an ˆ invariant of the cohomology algebra of S. ˆ Z) to There remains to show that, identifying the ﬁrst cohomology group H 1 (S, the one which we obtain for a homotopically equivalent primary Burniat surface, the class of Y remains the same. It suﬃces to show that, for each class γ ∈ H 4 (T, Z) = ∧4 (H 1 (T, Z)) the intersection product γ · Y is the same. But this number is the multiple of the ˆ Z), fundamental class of Sˆ yielding α ˆ ∗ (γ), and, identifying H 1 (T, Z) with H 1 (S, ∗ ˆ it is an invariant of the cohomology algebra H (S, Z), which is isomorphic to the one we obtain for a homotopically equivalent primary Burniat surface. The above claim implies that Sˆ → Y is a birational morphism and that Y ⊂ Z has cohomology class 2F1 + 2F2 + 2F3 . Thus KY = OY (Y ), and KY2 = (2F1 + 2F2 + 2F3 )3 = 48. On the other hand, since S is homotopically equivalent to a primary Burniat surface, we have that KS2 = 6, whence KS2ˆ = 6 · 23 = 48. Moreover, we have ˆ = q(S) ˆ + χ(S) ˆ − 1 = 3 + 8χ(S) − 1 = 10. pg (S) The short exact sequence 0 → OT → OT (Y ) → ωY → 0, induces a long exact cohomology sequence 0 → H 0 (T, OT ) → H 0 (T, OT (Y )) → H 0 (Y, ωY ) → → H 1 (T, OT ) → H 1 (T, OT (Y )) = 0, where the last equality holds since Y is an ample divisor on T . Moreover H 0 (T, OT ) ∼ = C, and H 1 (T, OT ) ∼ = C3 , and therefore ˆ pg (Y ) = h0 (Y, ωY ) = 10 = pg (S). Since |ωY | is base point free, and it has the same dimension as |ωSˆ |, this implies that Y has at most rational double points as singularities. This concludes the proof that S is a primary Burniat surface.

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4.1. The rationality of the moduli space of primary Burniat surfaces. This subsection is devoted to the proof of the following: Theorem 4.5. Let C be the connected component of the moduli space constituted by the primary Burniat surfaces (KS2 = 6). Then C is a rational 4-fold. Proof. The proof goes along similar lines as the one of the rationality of the moduli space of secondary Burniat surfaces with K 2 = 5, explained in [BC09b]. We have to divide a parameter space ∼ = C6 , parametrizing three pairs of lines of equations xi+2 = ai xi+1 , xi+2 = bi xi+1 by the action of (C∗ )2 , of S3 , of the (Z/2Z)3 generated by the transformations τi such that τi exchanges ai with bi , −1 and ﬁnally of the Cremona transformation (mapping ai to a−1 i , bi to bi ). Now, we can replace the action of S3 by the direct sum of two copies of the standard permutation representation (of the ai ’s and of the bi ’s). Moreover, we have the action of the subgroup (C∗ )2 ⊂ PGL(3, C) of diagonal matrices (C∗ )2 := {diag(t1 , t2 , t3 )|ti ∈ C∗ } : ai ?→ ai We set: λi :=

ti+1 ti+2 ,

thus

F i

ti+1 ti+1 , bi ?→ bi , i ∈ {1, 2, 3}. ti+2 ti+2

λi = 1 and our (C∗ )2 is the subgroup of (C∗ )3 ,

(C∗ )2 = {(λ1 , λ2 , λ3 )|

1

λi = 1}.

i

The invariants for the (Z/2Z)3 -action are: ui := ai bi , vi := ai + bi and (C∗ )3 acts by ui ?→ λ2i ui , vi ?→ λi vi . Claim 4.6. The invariants for the (C∗ )2 -action are 3

wi :=

1 ui , i = 1, 2, 3; v := vi . 2 vi i=1

Proof of the claim. Clearly the ﬁeld of (C∗ )3 -invariants is generated by the wi ’s, and we can replace the generators ui , vi (i = 1, 2, 3) by the generators wi , vi (i = 1, 2, 3). Now the exact sequence of algebraic groups 1 → (C∗ )2 → (C∗ )3 → C∗ → 1 F where (λ1 , λ2 , λ3 ) ?→ λ := i λi , shows that the projection (C∗ )3 → C∗ is the quotient map by the (C∗ )2 action. Since C(v1 , v2 , v3 ) is the function ﬁeld of (C∗ )3 , the ﬁeld of invariants for (C∗ )2 acting on C(v1 , v2 , v3 ) is C(v). Note that S3 acts on {w1 , w2 , w3 } by the permutationQrepresentation, whereas the Cremona transformation acts by wi ?→ wi , v ?→ Q uvii =: uv . In fact, the Cremona transformation sends ui to u−1 and vi to i

1 ai

+

1 bi

=

ai +bi ai bi

=

vi ui .

Since

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F F F = wi it follows that u = wi v 2 , thus v ?→ ( wi )−1 v −1 . The invariants for the Cremona transformations are therefore u v2

w1 , w2 , w3 , v +

v

1 F

wi

where the last element is S3 -invariant. Finally, the invariants for the action of S3 are: the elementary symmetric functions σ1 (w1 , w2 , w3 ), σ2 (w1 , w2 , w3 ), σ3 (w1 , w2 , w3 ), and v + v Q1 wi . Thus the ﬁeld of invariants is rational.

5. The Burniat surface with K 2 = 2 is a classical Campedelli surface The aim of this short last section is to illustrate how the Burniat surface with K 2 = 2 can be seen as a classical Campedelli surface (with fundamental group (Z/2Z)3 ). A classical Campedelli surface can be described as the tautological (Z/2Z)3 Galois-covering of P2 branched in seven lines. This means that each line {lα = 0} is set to correspond to a non zero element of the Galois group (Z/2Z)3 , and then, ∗ for each character χ ∈ (Z/2Z)3 , we consider the covering given (cf. [Par91]) by 1 wχ wχ" = lν wχ+χ" χ(ν)=χ" (ν)=1

> in the vector bundle whose sheaf of sections is χ∈(Z/2Z)3 ∗ OP2 (1). As we have seen before, the singular model X of a Burniat surface S with 2 KS2 = 2 (i.e., KX = 6) is the (Z/2Z)2 Galois covering branched in 9 lines having 4 points of type (1, 1, 1), whereas the minimal model S of a Burniat surface with KS2 = 2 is the smooth bidouble cover of a weak Del Pezzo surface Y DD of degree 2. Note that the strict transforms of the lines of D ⊂ P2 passing through 2 of the points P4 , P5 , P6 , P7 yield rational (−2)-curves on Y DD . There are six of them on Y DD , namely Di,j for 1 ≤ i ≤ 3, j ∈ {2, 3}. Contracting these six (−2) curves, we obtain a normal Del Pezzo surface Y D of degree 2 having six nodes, and with −KY " ample. Then the anticanonical map ϕ := ϕ|−KY " | : Y D → P2 is a ﬁnite double cover branched on a quartic curve, which has 6 nodes (since Y D has six nodes). But a plane quartic having 6 nodes has to be the union of four lines L1 , L2 , L3 , L4 in general position. Let S → Y DD be the bidouble cover branched in the Burniat conﬁguration yielding a minimal model of the Burniat surface with KS2 = 2. Then the preimages of the (−2)-curves Di,j for 1 ≤ i ≤ 3, j ∈ {2, 3} on Y are rational (−2) curves of S. Let now X D be the canonical model of S and consider the composition of the bidouble cover ψ : X D → Y D with ϕ. Since ψ branches on the image Δ of D1,1 + D2,1 + D3,1 in Y D (the other 6 lines being contracted), we see that the branch divisor of ϕ ◦ ψ consists of Q := L1 + L2 + L3 + L4 and the image of Δ in P2 .

Burniat surfaces I

75

Looking at the conﬁguration of the lines (cf. ﬁgure 1), we see that D1,1 intersects D2,2 and D2,3 , that D2,1 intersects D3,2 and D3,3 , and that D3,1 intersects D1,2 and D1,3 . Hence the image of Di,1 under ϕ has to intersect two nodes of the plane quartic Q := L1 + L2 + L3 + L4 , which implies that, denoting the image of Di,1 under ϕ by LDi , the branch divisor of the (Z/2Z)3 Galois-covering ϕ ◦ ψ : X → P2 is a conﬁguration of seven lines L1 + L2 + L3 + L4 + LD1 + LD2 + LD3 , where L1 , L2 , L3 , L4 are four lines in general position, i.e., form a complete quadrilateral, and LD1 , LD2 , LD3 are the three diagonals. The covering ϕ◦ψ : X → P2 is a Galois covering with Galois group (Z/2Z)3 . In fact we already have as covering transformations the elements of the Galois group GDD := (Z/2Z)2 of ψ. Moreover the involution i : Y D → Y D can be lifted to X D since i leaves the individual branch curves invariant (as they are inverse image of the diagonals of the quadrilateral), and also the line bundles associated to the covering of Y DD (Y DD is simply connected, whence division by 2 is unique in Pic(Y DD )). To show that the covering is the tautological one it suﬃces to verify that for each non trivial element of the Galois group its ﬁxed divisor is exactly the inverse image of one of the 7 lines in P2 . We omit further details since they are contained in the article [Ku04] by Kulikov. The idea there is simply to take the tautological cover and observe that it factors as a bidouble cover of Y D branched on the inverse image of the diagonals, each splitting into the divisor corresponding to the line Di,1 and the divisor corresponding to Ei+2 . Whence Kulikov veriﬁes that one gets in this way the Burniat surface with K 2 = 2. Remark 5.1. There are other interesting (Z/2Z)3 -Galois covers of the plane branched on the seven lines L1 , L2 , L3 , L4 , LD1 , LD2 , LD3 . One such is the ﬁbre product Z of the standard bidouble cover P2 → P2 branched on the diagonals LD1 , LD2 , LD3 with the double covering Y D branched on L1 , L2 , L3 , L4 . This gives Z as a double plane branched on four conics touching in 12 points. Z is a surface with KZ2 = 2, pg (Z) = 3, whose singularities are precisely 12 points of type A3 .

References [Arm65]

Armstrong, M. A., On the fundamental group of an orbit space. Proc. Cambridge Philos. Soc. 61 639–646 (1965).

[Arm68]

Armstrong, M. A., The fundamental group of the orbit space of a discontinuous group. Proc. Cambridge Philos. Soc. 64 299–301 (1968).

[BCGP09] Bauer, I., Catanese, F., Grunewald, F., Pignatelli, R. Quotients of a product of curves by a ﬁnite group and their fundamental groups. arXiv:0809.3420. [BC09a]

Bauer, I., Catanese, F. The moduli space of Keum - Naie surfaces arXiv: 0909.1733.

[BC09b]

Bauer, I., Catanese, F. Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space. arXiv:0911.1466.

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[Brie68]

Brieskorn, E. Die Auﬂ¨ osung der rationalen Singularit¨ aten holomorpher Abbildungen. Math. Ann. 178 (1968) 255–270.

[Brie71]

Brieskorn, E. Singular elements of semi-simple algebraic groups. Actes du Congr´es International des Math´ematiciens (Nice, 1970), Tome 2, pp. 279–284. Gauthier-Villars, Paris, 1971.

[Bu66]

Burniat, P. Sur les surfaces de genre P12 > 1. Ann. Mat. Pura Appl. (4) 71 (1966) 1–24.

[Cat84]

Catanese, F. On the moduli spaces of surfaces of general type. J. Diﬀerential Geom. 19 (1984), no. 2, 483–515.

[Cat99]

Catanese, F. Singular bidouble covers and the construction of interesting algebraic surfaces. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97– 120, Contemp. Math. 241, Amer. Math. Soc., Providence, RI, 1999.

[Cat08]

Catanese, F. Diﬀerentiable and deformation type of algebraic surfaces, real and symplectic structures. Symplectic 4-manifolds and algebraic surfaces, 55– 167, Lecture Notes in Math., 1938, Springer, Berlin, 2008.

[In94]

Inoue, M. Some new surfaces of general type. Tokyo J. Math. 17 (1994), no. 2, 295–319.

[Ku04]

Kulikov, V., S. Old examples and a new example of surfaces of general type with pg = 0. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 5, 123–170; translation in Izv. Math. 68 (2004), no. 5, 965–1008.

[MLP01]

Mendes Lopes, M., Pardini, R. A connected component of the moduli space of surfaces with pg = 0. Topology 40 (2001), no. 5, 977–991.

[Miy77]

Miyaoka, Y. On numerical Campedelli surfaces. In: Complex analysis and algebraic geometry, pp. 113–118. Iwanami Shoten, Tokyo, 1977.

[Par91]

Pardini, R. Abelian covers of algebraic varieties. J. Reine Angew. Math. 417 (1991), 191–213.

[Pet77]

Peters, C. A. M. On certain examples of surfaces with pg = 0 due to Burniat. Nagoya Math. J. 66 (1977), 109–119.

[Rei]

Reid, M. Surfaces with pg = 0, K 2 = 2. Preprint available at http:// www.warwick.ac.uk/ masda/surf/.

[Rei79]

Reid, M.: π1 for surfaces with small K 2 . In: Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), 534–544. Lecture Notes in Math. 732, Springer, Berlin (1979).

[Tju70]

Tjurina, G. N. Resolution of singularities of ﬂat deformations of double rational points. Funkcional. Anal. i Prilozen. 4 (1970) no. 1, 77–83.

I. Bauer, Lehrstuhl Mathematik VIII, Mathematisches Institut der Universit¨ at Bayreuth, NW II, Universit¨ atsstr. 30, 95447 Bayreuth E-mail: [email protected] F. Catanese, Lehrstuhl Mathematik VIII, Mathematisches Institut der Universit¨ at Bayreuth, NW II, Universit¨ atsstr. 30, 95447 Bayreuth E-mail: [email protected]

Minimal models, ﬂips and ﬁnite generation: a tribute to V.V. Shokurov and Y.-T. Siu C. Birkar, M. P˘aun

Abstract. In this paper, we give a proof of the existence of log minimal models for klt pairs (X/Z, B) with B big/Z. This then implies existence of klt log ﬂips, ﬁnite generation of klt log canonical rings, and most of the other results of the Birkar-CasciniHacon-McKernan paper [3].

1. Introduction We consider pairs (X/Z, B) where B is an R-boundary and X → Z is a projective morphism of normal quasi-projective varieties over an algebraically closed ﬁeld k of characteristic zero. We call a pair (X/Z, B) eﬀective if there is an R-divisor M ≥ 0 such that KX + B ≡ M/Z where ≡ denotes numerical equivalence. Theorem 1.1. Let (X/Z, B) be a klt pair where B is big/Z. Then, (1) if KX + B is pseudo-eﬀective/Z, then (X/Z, B) has a log minimal model, (2) if KX + B is not pseudo-eﬀective/Z, then (X/Z, B) has a Mori ﬁbre space. Corollary 1.2 (Log ﬂips). Log ﬂips exist for klt (hence Q-factorial dlt) pairs. Corollary 1.3 (Finite generation). Let (X/Z, B) be a klt pair where B is a Qdivisor and f : X → Z the given morphism. Then, the log canonical sheaf . R(X/Z, B) := f∗ OX (\m(KX + B)L) m≥0

is a ﬁnitely generated OZ -algebra. The proof of the above theorem is divided into two independent parts. First we have Theorem 1.4 (Log minimal models). Assume (1) of Theorem 1.1 in dimension d − 1 and let (X/Z, B) be a klt pair of dimension d where B is big/Z. If (X/Z, B) is eﬀective, then it has a log minimal model. A proof of this theorem is given in section 2 based on ideas in [2][3]. Second we have Theorem 1.5 (Nonvanishing). Let (X/Z, B) be a klt pair where B is big/Z. If KX + B is pseudo-eﬀective/Z, then (X/Z, B) is eﬀective.

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A complete proof of this theorem is given in section 3; it is mainly based on the arguments in [26] (see [27] and [34] as well). We give a rough comparison of the proof of this theorem with the proof of the corresponding theorem in [3], that is [3], Theorem D. One crucial feature of the proof of Theorem 1.5 is that it does not use the log minimal model program. The proof goes as follows: (a) We ﬁrst assume that Z is a point, and using Zariski type decompositions one can create lc centres and pass to plt pairs, more precisely, by going on a suﬃciently high resolution we can replace (X/Z, B) by a plt pair (X/Z, B + S) where S is a smooth prime divisor, KX + B + S|S is pseudo-eﬀective, B is big and its components do not intersect. (b) By induction, the R-bundle KX + B + S|S has an R-section, say T , which can be assumed to be singular enough for the extension purposes. (c) Diophantine approximation of the couple (B, T ): we can ﬁnd couples (Bi , Ti ) with rational coeﬃcients and suﬃciently close to (B, T ) (in a very precise sense) such that M KX + B + S = ri (KX + Bi + S) for certain real numbers ri ∈ [0, 1] and such that all the pairs (X/Z, Bi + S) are plt and each KX + Bi + S|S is numerically equivalent with Ti . Moreover, one can improve this to KX + Bi + S|S ∼Q Ti ≥ 0. (d) Using the invariance of plurigenera techniques, one can lift this to KX + Bi + S ∼Q Mi ≥ 0 and then a relation KX + B + S ≡ M ≥ 0. (e) Finally, we get the theorem in the general case, i.e., when Z is not a point, using positivity properties of direct image sheaves and another application of extension theorems. In contrast, the log minimal model program is an important ingredient of the proof of [3], Theorem D, which proceeds as follows: (a’) This step is the same as (a) above. (b’) By running the log minimal model program appropriately and using induction on ﬁniteness of log minimal models and termination with scaling, one constructs a model Y birational to X such that KY + BY + SY |SY is nef, where KY + BY + SY is the pushdown of KX + B + S. Moreover, by Diophantine approximation, we can ﬁnd%boundaries Bi with rational coeﬃcients and suﬃciently close to B such that B = ri Bi for certain real numbers ri ∈ [0, 1] and such that each KY +Bi,Y +SY is plt and KY +Bi,Y +SY |SY is nef. (c’) By applying induction and the base point free theorem one gets KY + Bi,Y + SY |SY ∼Q Ni ≥ 0. (d’) The Kawamata-Viehweg vanishing theorem now gives KY + Bi,Y + SY ∼Q Mi,Y ≥ 0 from which we easily get a relation KX + B + S ≡ M ≥ 0. (e’) Finally, we get the theorem in the general case, i.e., when Z is not a point, by restricting to the generic ﬁbre and applying induction.

2. Log minimal models In this section we prove Theorem 1.4 (cf. [3], Theorems A, B, C, E). The results in this section are also implicitly or explicitly proved in [3]. We hope that this

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section also helps the reader to read [3]. Preliminaries. Let k be an algebraically closed ﬁeld of characteristic zero ﬁxed % throughout this section. When we write an R-divisor D as D = di Di (or similar notation) we mean that Di are distinct prime divisors. The norm ||D|| is deﬁned as max{|di |}. For a birational map φ : X ''( Y and an R-divisor D on X we often use DY to mean the birational transform of D, unless speciﬁed otherwise. A pair (X/Z, B) consists of normal quasi-projective varieties X, Z over k, an R-divisor B on X with coeﬃcients in [0, 1] such that KX + B is R-Cartier, and a projective morphism X → Z. For a prime divisor E on some birational model of X with a nonempty centre on X, a(E, X, B) denotes the log discrepancy. An R-divisor D on X is called pseudo-eﬀective/Z if it is the limit of eﬀective R-divisors up to numerical equivalence/Z, i.e., for any ample/Z R-divisor A and real number a > 0, D + aA is big/Z. A pair (X/Z, B) is called eﬀective if there is an R-divisor M ≥ 0 such that KX + B ≡ M/Z; in this case, we call (X/Z, B, M ) a triple. By a log resolution of a triple (X/Z, B, M ) we mean a log resolution of (X, Supp(B + M )). A triple (X/Z, B, M ) is log smooth if (X, Supp(B + M )) is log smooth. When we refer to a triple as being lc, dlt, etc, we mean that the underlying pair (X/Z, B) has such properties. For a triple (X/Z, B, M ), deﬁne θ(X/Z, B, M ) := #{i | mi K= 0 and bi K= 1}, % where B = bi Di and M = mi Di . Let (X/Z, B) be an lc pair. By a log ﬂip/Z we mean the ﬂip of a KX + B-negative extremal ﬂipping contraction/Z, and by a pl ﬂip/Z we mean a log ﬂip/Z when (X/Z, B) is Q-factorial dlt and the log ﬂip is also an S-ﬂip for some component S of \BL, i.e., S is numerically negative on the ﬂipping contraction. A sequence of log ﬂips/Z starting with (X/Z, B) is a sequence Xi ''( Xi+1 /Zi in which Xi → Zi ← Xi+1 is a KXi + Bi -ﬂip/Z, Bi is the birational transform of B1 on X1 , and (X1 /Z, B1 ) = (X/Z, B). %

Deﬁnition 2.1 (Log minimal models and Mori ﬁbre spaces). Let (X/Z, B) be a dlt pair, (Y /Z, BY ) a Q-factorial dlt pair, φ : X ''( Y /Z a birational map such that φ−1 does not contract divisors, and BY = φ∗ B. (1) We say that (Y /Z, BY ) is a nef model of (X/Z, B) if KY + BY is nef/Z. We say that (Y /Z, BY ) is a log minimal model of (X/Z, B) if in addition a(D, X, B) < a(D, Y, BY ) for any prime divisor D on X which is exceptional/Y . (2) Let (Y /Z, BY ) be a log minimal model of (X/Z, B) such that KY + BY is semi-ample/Z so that there is a contraction f : Y → S/Z and an ample/Z Rdivisor H on S such that KY + BY ∼R f ∗ H/Z. We call S the log canonical model of (X/Z, B), which is unique up to isomorphism/Z. (3) On the other hand, we say that (Y /Z, BY ) is a Mori ﬁbre space of (X/Z, B) if there is a KY + BY -negative extremal contraction Y → T /Z such that dim T <

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dim Y , and if

a(D, X, B) ≤ a(D, Y, BY )

for any prime divisor D on birational models of X, with strict inequality for any prime divisor D on X which is exceptional/Y . Note that in [2], it is not assumed that φ−1 does not contract divisors. However, since in this paper we are mainly concerned with constructing models for klt pairs, in that case our deﬁnition here is equivalent to that of [2]. Lemma 2.2. Let (X/Z, B+C) be a Q-factorial lc pair where B, C ≥ 0, KX +B+C is nef/Z, and (X/Z, B) is dlt. Then, either KX + B is also nef/Z or there is an extremal ray R/Z such that (KX + B) · R < 0, (KX + B + λC) · R = 0, and KX + B + λC is nef/Z where λ := inf{t ≥ 0 | KX + B + tC is nef/Z}. Proof. This is [2], Lemma 3.1. We give the proof here for convenience. Suppose that KX + B is not nef/Z and let {Ri }i∈I be the set of (KX + B)negative extremal rays/Z and Γi an extremal curve of Ri ([31], Deﬁnition 1). Let µ := sup{µi } where −(KX + B) · Γi . µi := C · Γi Obviously, λ = µ and µ ∈ (0, 1]. It is enough to prove that µ = µl for some l. By [31], Proposition 1, there are positive real numbers r1 , . . . , rs and a positive integer m (all independent of i) such that (KX + B) · Γi =

s M rj ni,j j=1

m

,

where −2(dim X)m ≤ ni,j ∈ Z. On the other hand, by [29], First Main Theorem 6.2 and Remark 6.4, we can write KX + B + C =

t M

rkD (KX + Δk ),

k=1

where r1D , · · · , rtD are positive real numbers such that for any k we have: (X/Z, Δk ) is lc with Δk being rational, and (KX + Δk ) · Γi ≥ 0 for any i. Therefore, there is a positive integer mD (independent of i) such that (KX + B + C) · Γi =

t M rkD nDi,k k=1

where 0 ≤ nDi,k ∈ Z. The set {ni,j }i,j is ﬁnite. Moreover,

mD

,

% m k rkD nDi,k C · Γi (KX + B + C) · Γi 1 + 1. = = +1=− D% µi −(KX + B) · Γi −(KX + B) · Γi m j rj ni,j

Thus, inf{ µ1i } =

1 µl

for some l and so µ = µl .

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Deﬁnition 2.3 (LMMP with scaling). Let (X/Z, B + C) be an lc pair such that KX + B + C is nef/Z, B ≥ 0, and C ≥ 0 is R-Cartier. Suppose that either KX + B is nef/Z or there is an extremal ray R/Z such that (KX + B) · R < 0, (KX + B + λ1 C) · R = 0, and KX + B + λ1 C is nef/Z where λ1 := inf{t ≥ 0 | KX + B + tC is nef/Z}. When (X/Z, B) is Q-factorial dlt, the last sentence follows from Lemma 2.2. If R deﬁnes a Mori ﬁbre structure, we stop. Otherwise assume that R gives a divisorial contraction or a log ﬂip X ''( X D . We can now consider (X D /Z, B D + λ1 C D ) where B D + λ1 C D is the birational transform of B + λ1 C and continue the argument. That is, suppose that either KX " + B D is nef/Z or there is an extremal ray RD /Z such that (KX " + B D ) · RD < 0, (KX " + B D + λ2 C D ) · RD = 0, and KX " + B D + λ2 C D is nef/Z where λ2 := inf{t ≥ 0 | KX " + B D + tC D is nef/Z}. By continuing this process, we obtain a special kind of LMMP/Z which is called the LMMP/Z on KX + B with scaling of C; note that it is not unique. This kind of LMMP was ﬁrst used by Shokurov [28]. When we refer to termination with scaling we mean termination of such an LMMP. Special termination with scaling means termination near \BL of any sequence of log ﬂips/Z with scaling of C, i.e., after ﬁnitely many steps, the locus of the extremal rays in the process does not intersect Supp \BL. When we have an lc pair (X/Z, B), we can always ﬁnd an ample/Z R-Cartier divisor C ≥ 0 such that KX + B + C is lc and nef/Z, so we can run the LMMP/Z with scaling assuming that all the necessary ingredients exist, e.g., extremal rays and log ﬂips. Finiteness of models. (P) Let X → Z be a projective morphism of normal quasi-projective varieties, A ≥ 0 a Q-divisor on X, and V a rational (i.e., with a basis consisting of rational divisors) ﬁnite dimensional aﬃne subspace of the space of R-Weil divisors on X. Deﬁne LA (V ) = {B | 0 ≤ (B − A) ∈ V, and (X/Z, B) is lc}. By [28, 1.3.2], [29], LA (V ) is a rational polytope (i.e., a polytope with rational vertices) inside the rational aﬃne space A + V . Remark 2.4. With the setting as in (P) above, assume that A is big/Z. Let B ∈ LA (V ) such that (X/Z, B) is klt. We can write A ∼R AD + G/Z where AD ≥ 0 is an ample/Z Q-divisor and G ≥ 0 is also a Q-divisor. Then, there is a suﬃciently small rational number r > 0 such that (X/Z, ΔB := B − rA + rAD + rG) is klt. Note that

KX + ΔB ∼R KX + B/Z.

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Moreover, there is a neighborhood of B in LA (V ) such that for any B D in that neighborhood (X/Z, ΔB " := B D − rA + rAD + rG) is klt. In particular, if C ⊆ LA (V ) is a rational polytope containing B, then perhaps after shrinking C (but preserving its dimension) we can assume that D := {ΔB " | B D ∈ C} is a rational polytope of klt boundaries in LlA" (W ) where W is the rational aﬃne space V + (1 − r)A + rG. The point is that we can change A and get an ample part rAD in the boundary. So, when we are concerned with a problem locally around B we feel free to assume that A is actually ample by replacing it with rAD . Lemma 2.5. With the setting as in (P) above, assume that A is big/Z, (X/Z, B) is klt of dimension d, and KX + B is nef/Z where B ∈ LA (V ). Then, there is an r > 0 (depending on X → Z, V, A, B) such that if R is a KX + B D -negative extremal ray/Z for some B D ∈ LA (V ) with ||B − B D || < r then (KX + B) · R = 0. Proof. This is proved in [31], Corollary 9, in a more general situation. Since B is big/Z and KX + B is nef/Z, the base point free theorem (cf. [13], Theorem 5.2.1) implies that KX + B is semi-ample/Z, hence there is a contraction f : X → S/Z ∗ and an ample/Z % R-Cartier divisor H on S such that KX + B ∼R f H/Z. We can write H ∼R ai Hi /Z where ai > 0 and the Hi are ample/Z Cartier divisors on S. Therefore, there is a δ > 0 such that for any curve C/Z in X either (KX +B)·C = 0 or (KX + B) · C > δ. Now let C ⊂ LA (V ) be a rational polytope of maximal dimension which contains an open neighborhood of B in LA (V ) and such that (X/Z, B D ) is klt for any B D ∈ C. Pick B D ∈ C and let B DD be the point on the boundary of C such that B D belongs to the line segment determined by B, B DD . Assume that R is a KX + B D -negative extremal ray/Z such that (KX + B) · R > 0. Then (KX + B DD ) · R < 0 and there is a rational curve Γ in R such that (KX + B DD ) · Γ ≥ −2d. Since (KX + B D ) · Γ < 0, (B D − B) · Γ = (KX + B D ) · Γ − (KX + B) · Γ < −δ. Now, ||B DD − B|| > α for some α > 0 independent of B DD . Thus, if ||B − B D || is too small, then (B DD − B D ) · Γ is too negative and we cannot have (KX + B DD ) · Γ = (KX + B D ) · Γ + (B DD − B D ) · Γ ≥ −2d. So, we get a contradiction. Theorem 2.6. Assume (1) of Theorem 1.1 in dimension d. With the setting as in (P) above, assume that A is big/Z. Let C ⊆ LA (V ) be a rational polytope such that (X/Z, B) is klt for any B ∈ C where dim X = d. Then, there are ﬁnitely many birational maps φi : X ''( Yi /Z such that for any B ∈ C with KX + B pseudo-eﬀective/Z, there is an i such that (Yi /Z, BYi ) is a log minimal model of (X/Z, B).

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Proof. Remember that as usual BYi is the birational transform of B. We do induction on the dimension of C. In particular, we may assume that the dimension of C is positive. We may proceed locally, so ﬁx B ∈ C. If KX + B is not pseudoeﬀective/Z then the same holds in a neighborhood of B inside C, so we may assume that KX + B is pseudo-eﬀective/Z. By assumption, (X/Z, B) has a log minimal model (Y /Z, BY ). Moreover, the polytope C determines a rational polytope CY of R-divisors on Y by taking birational transforms of elements of C. If we shrink C around B we can assume that the inequality in (1) of Deﬁnition 2.1 is satisﬁed for every B D ∈ C, that is, a(D, X, B D ) < a(D, Y, BYD ) for any prime divisor D ⊂ X contracted/Y . Moreover, for each B D ∈ C, a log minimal model of (Y /Z, BYD ) is also a log minimal model of (X/Z, B D ). Therefore, we can replace (X/Z, B) by (Y /Z, BY ) and assume from now on that (X/Z, B) is a log minimal model of itself, in particular, KX + B is nef/Z. Since B is big/Z, by the base point free theorem, KX + B is semi-ample/Z so there is a contraction f : X → S/Z such that KX + B ∼R f ∗ H/Z for some ample/Z R-divisor H on S. Now by induction on the dimension of C, we may assume that there are ﬁnitely many birational maps ψj : X ''( Yj /S such that for any B DD on the boundary of C with KX + B DD pseudo-eﬀective/S, there is a j such that (Yj /S, BYDDj ) is a log minimal model of (X/S, B DD ). By Lemma 2.5, there is a suﬃciently small r > 0 such that for any B D ∈ C with ||B − B D || < r, any j, and any KYj + BYD j -negative extremal ray R/Z we have the equality (KYj + BYj ) · R = 0. Note that all the pairs (Yj /Z, BYj ) are klt and KYj + BYj ∼R 0/S and nef/Z because KX + B ∼R 0/S. Pick B D ∈ C with 0 < ||B − B D || < r such that KX + B D is pseudo-eﬀective/Z, and let B DD be the unique point on the boundary of C such that B D belongs to the line segment given by B and B DD . Since KX + B ∼R 0/S, there is some t > 0 such that KX + B DD = KX + B + B DD − B ∼R B DD − B = t(B D − B) ∼R t(KX + B D )/S, hence KX + B DD is pseudo-eﬀective/S, and (Yj /S, BYDDj ) is a log minimal model of (X/S, B DD ) for some j. Moreover, (Yj /S, BYD j ) is a log minimal model of (X/S, B D ). Furthermore, (Yj /Z, BYD j ) is a log minimal model of (X/Z, B D ) because any KYj + BYD j -negative extremal ray R/Z would be over S by the choice of r. Finally, we just need to shrink C around B appropriately. Termination with scaling. Remark 2.7. Assume (1) of Theorem 1.1 in dimension d and let (X/Z, Δ + Γ) be a klt pair of dimension d with Δ ≥ 0 big/Z, Γ ≥ 0 being R-Cartier, and KX + Δ + Γ ≡ 0/Z. Assume that KX + Δ is pseudo-eﬀective/Z and let X D /Z be its lc model and X ''( X D /Z the induced rational map. Let Y → Z be a small Qfactorialisation of X which exists by Lemma 2.11 and Remark 2.12, and let ΔY , ΓY denote birational transforms as usual. Now run the LMMP/Z on KY + ΔY with

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scaling of ΓY . If the LMMP terminates with a log minimal model (Y D /Z, ΔY " ), then Y D → Z factors through X D because X D /Z is the lc model of (X/Z, Δ) as well as of (Y /Z, ΔY ). We call the birational map Y ''( Y D a Q-factorial lift of X ''( X D . By construction, Y ''( Y D is an isomorphism or else it is decomposed into a ﬁnite sequence Yi ''( Yi+1 /Zi of divisorial contractions and log ﬂips/Z such that KYi + ΔYi + ΓYi ≡ 0/Zi and ΓYi is ample/Zi . If X ''( X D is an isomorphism in codimension one (e.g., a log ﬂip), then only log ﬂips can occur in the sequence Yi ''( Yi+1 /Zi . One can use this construction to lift a sequence of log ﬂips in the non-Q-factorial case to a sequence of log ﬂips in the Q-factorial case. Theorem 2.8. Assume (1) of Theorem 1.1 in dimension d and let (X/Z, B +C) be a klt pair of dimension d where B ≥ 0 is big/Z, C ≥ 0 is R-Cartier, and KX +B+C is nef/Z. Then, any LMMP/Z on KX + B with scaling of C terminates. Proof. Assume that we are given an LMMP/Z on KX + B with scaling of C and assume that we get an inﬁnite sequence Xi ''( Xi+1 /Zi of log ﬂips/Z. We may assume that X = X1 . Let λi be as in Deﬁnition 2.3 and put λ = lim λi . So, by deﬁnition, KXi + Bi + λi Ci is nef/Z and numerically zero over Zi , where Bi and Ci are the birational transforms of B and C respectively. By Remark 2.7, we can assume that all the Xi are Q-factorial by lifting the sequence to the Q-factorial situation. Let H1 , . . . , Hm be general ample/Z Cartier divisors on X which generate the space N 1 (X/Z). Since B is big/Z, we may assume that B − r(H1 + · · · + Hm ) ≥ 0 for some rational number r > 0 (see Remark 2.4). Put A = 2l (H1 + · · · + Hm ). Let V be the space generated by the components of B + C, and let C ⊂ LA (V ) be a rational polytope of maximal dimension containing neighborhoods of B and B + C such that (X/Z, B D ) is klt for any B D ∈ C. Moreover, we can choose C such that % for each i there is an ample/Z Q-divisor Gi = gi,j Hi,j on Xi with suﬃciently small coeﬃcients, where Hi,j on Xi is the birational transform of Hj , such that Δi , the birational transform of Bi + Gi + λi Ci on X, belongs to C and such that (Xi , Bi + Gi + λi Ci ) is an lc model of (X, Δi ). On the other hand, by Theorem 2.6, there are ﬁnitely many birational maps φl : X ''( Yl /Z such that for any B D ∈ C with KX + B D pseudo-eﬀective/Z, there is an l such that (Yl /Z, BYD l ) is a log minimal model of (X/Z, B D ). Since KXi + Bi + Gi + λi Ci is ample/Z and since the log canonical model is unique, for each is an isomorphism, where φi,j is the birational i, there is some l such that φ1,i φ−1 l map Xi ''( Xj . Therefore, there exist an l and an inﬁnite set I ⊆ N such that is an isomorphism for any i ∈ I. This in turn implies that φi,j is an φ1,i φ−1 l isomorphism for any i, j ∈ I. This is not possible as any log ﬂip increases some log discrepancies. Note that existence of klt log ﬂips in dimension d follows from the assumptions of Theorem 2.8 (see the proof of Corollary 1.2). So, if X is Q-factorial, under the assumptions of the theorem, we can actually run an LMMP/Z on KX + B with scaling of C by Lemma 2.2.

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Theorem 2.9. Assume (1) of Theorem 1.1 in dimension d−1 and let (X/Z, B+C) be a Q-factorial dlt pair of dimension d where B − \BL − A ≥ 0 for some ample/Z R-divisor A ≥ 0, and C ≥ 0. Assume that (Y /Z, BY + CY ) is a log minimal model of (X/Z, B + C). Then, the special termination holds for any LMMP/Z on KY + BY with scaling of CY . Proof. Suppose that we have an LMMP/Z on KY + BY with scaling of CY producing a sequence Yi ''( Yi+1 /Zi of log ﬂips/Z. Let S be a component of \BL and let SY and SYi be its birational transforms on Y and Yi respectively. Let λi be as in Deﬁnition 2.3 for the sequence Yi ''( Yi+1 /Zi . That is, KYi + BYi + λi CYi is nef/Z but numerically zero over Zi , and CYi is ample/Zi . First suppose that λi = 1 for every i. Since B − \BL − A ≥ 0 and since A is ample/Z, using a simple perturbation of coeﬃcients we can write B + C ∼R AD + B D + C D /Z, where for a suﬃciently small rational number r > 0 AD ∼R A + rC + r \B − SL is ample/Z, and B D = B − A − r \B − SL ≥ 0,

C D = (1 − r)C,

\B D L = \AD + B D + C D L = S. Moreover, perhaps after another small perturbation we may assume that AD is a Q-divisor and that the pairs (X/Z, AD + B D + C D ) and (Y /Z, ADY + BYD + CYD ) are plt, and that the above LMMP/Z on KY + BY with scaling of CY is an LMMP/Z on KY + ADY + BYD with scaling of CYD . K 0, otherwise there is nothing to prove. Following some Assume that SY1 = standard arguments (cf. [11]), we may assume that the birational maps SYi+1 ''( SYi do not contract divisors. Since ADYi is the pushdown of an ample/Z divisor, ADYi |SYi is big/Z. Moreover, if Ti is the normalisation of the image of SYi in Zi , then (KYi + ADYi + BYD i + λi CYD i )|SYi ∼R 0/Ti . Furthermore, by taking Q-factorial lifts of the maps SYi ''( SYi+1 as in Remark 2.7 and applying Theorem 2.8 in dimension d − 1, we deduce that SYi ''( SYi+1 are isomorphisms for i D 0, hence the log ﬂips in the sequence Yi ''( Yi+1 /Zi do not intersect SYi for i D 0. Now assume that we have λi < 1 for some i. Then, \B + λi CL = \BL for any i D 0. So, we may assume that \B + CL = \BL. Since B − \BL − A ≥ 0 and since A is ample/Z, similarly to the above, we can write B ∼R AD + B D /Z such that AD ≥ 0 is an ample/Z Q-divisor, B D ≥ 0,

\B D L = \AD + B D + CL = S

86 and

C. Birkar, M. P˘ aun

(X/Z, AD + B D + C) and (Y /Z, ADY + BYD + CY )

are plt. The rest goes as before by restricting to the birational transforms of SY . Pl ﬂips. We need an important result of Hacon-McKernan [13], which in turn is based on important works of Shokurov [30][28], Siu [32] and Kawamata [17]. Theorem 2.10. Assume (1) of Theorem 1.1 in dimension d − 1. Then, pl ﬂips exist in dimension d. Proof. By Theorem 2.8 and Corollary 1.2 in dimension d − 1, [13, Assumption 5.2.3] is satisﬁed in dimension d − 1. Note that Corollary 1.2 in dimension d − 1 easily follows from (1) of Theorem 1.1 in dimension d−1 (see the proof of Corollary 1.2). Now [13, Theorem 5.4.25, proof of Lemma 5.4.26] implies the result. Log minimal models. Lemma 2.11. Assume (1) of Theorem 1.1 in dimension d − 1. Let (X/Z, B) be a klt pair of dimension d and let {Di }i∈I be a ﬁnite set of exceptional/X prime divisors (on birational models of X) such that the log discrepancy a(Di , X, B) ≤ 1. Then, there is a Q-factorial klt pair (Y /X, BY ) such that (1) Y → X is birational and KY + BY is the crepant pullback of KX + B, (2) the set of exceptional/X prime divisors of Y is exactly {Di }i∈I . Proof. Let f : W → X be a log resolution of (X/Z, B) and let {Ej }j∈J be the set of prime exceptional divisors of f . We can assume that for some J D ⊆ J, {Ej }j∈J " = {Di }i∈I . Since f is birational, there is an ample/X Q-divisor H ≥ 0 on W whose support is irreducible, smooth, and distinct from the birational transform of the components of B, and an R-divisor G ≥ 0 such that H + G ∼R 0/X. Moreover, there is an r > 0 such that (X/Z, B + rf∗ H + rf∗ G) is klt. Now deﬁne M KW + B W := f ∗ (KX + B + rf∗ H + rf∗ G) + a(Ej , X, B + rf∗ H + rf∗ G)Ej , j ∈J / "

for which obviously there is an exceptional/X R-divisor M W ≥ 0 such that KW + B W ∼R M W /X and θ(W/X, B W , M W ) = 0. E 0 Note that B W − B W ≥ rH. By Theorem 2.10, we can run an LMMP/X on KW + B W with scaling of a suitable ample/X R-divisor, and using the special termination of Theorem 2.9 we get a log minimal model of (W/X, B W ) which we may denote by (Y /X, B Y ). Note that here we only need pl ﬂips to run the LMMP/X because any extremal ray in the process intersects some component of E 0 B W negatively. The exceptional divisor Ej is contracted/Y exactly when j ∈ / J D . By taking KY + BY to be the crepant pullback of KX + B we get the result.

Minimal models, ﬂips and ﬁnite generation

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Remark 2.12. In the lemma, if we assume (1) of Theorem 1.1 in dimension d, then we can prove the lemma without appealing to Theorem 2.9 in lower dimension. Indeed, we need Theorem 2.9 to get a log minimal model of (W/X, B W ). But that existence directly follows from (1) of Theorem 1.1 in dimension d. Note that (W/X, B W ) is not klt, but B W has an ample/Z component, hence perturbing the coeﬃcients reduces the problem to the klt case. Proof (of Theorem 1.4). We closely follow the proof of [2], Theorem 1.3. Remember that the assumptions imply that pl ﬂips exist in dimension d by Theorem 2.10 and that the special termination holds as in Theorem 2.9. Step 1. Since B is big/Z, by perturbing the coeﬃcients we can assume that it has a general ample/Z component which is not a component of M (see Remark 2.4). By taking a log resolution we can further assume that the triple (X/Z, B, M ) is log smooth. To construct log minimal models in this situation we need to pass to a more general setting. Let W be the set of triples (X/Z, B, M ) which satisfy (1) (X/Z, B) is dlt of dimension d and (X/Z, B, M ) is log smooth, (2) (X/Z, B) does not have a log minimal model, (3) B has a component which is ample/Z but it is not a component of \BL nor a component of M . Obviously, it is enough to prove that W is empty. Assume otherwise and choose (X/Z, B, M ) ∈ W with minimal θ(X/Z, B, M ). If θ(X/Z, B, M ) = 0, then either M = 0, in which case we already have a log minimal model, or by running the LMMP/Z on KX + B with scaling of a suitable ample/Z R-divisor we get a log minimal model, because by the special termination of Theorem 2.9, ﬂips and divisorial contractions will not intersect Supp \BL ⊇ Supp M after ﬁnitely many steps. This is a contradiction. Note that we need only pl ﬂips here, which exist by Theorem 2.10. We may then assume that θ(X/Z, B, M ) > 0. % % D Step 2. Notation: for an R-divisor D = di Di we deﬁne D≤1 := di Di in which dDi = min{di , 1}. Now put E 0 α := min{t > 0 | (B + tM )≤1 = K \BL }. In particular, (B + αM )≤1 = B + C for some C ≥ 0 supported in Supp M , and αM = C+M D where M D is supported in Supp \BL. Thus, outside Supp \BL we have C = αM . The pair (X/Z, B +C) is obviously log smooth and (X/Z, B +C, M +C) is a triple which satisﬁes (1) and (3) above. By construction θ(X/Z, B + C, M + C) < θ(X/Z, B, M ), so (X/Z, B + C, M + C) ∈ / W. Therefore, (X/Z, B + C) has a log minimal model, say (Y /Z, BY + CY ). By deﬁnition, KY + BY + CY is nef/Z. Step 3. Now run the LMMP/Z on KY + BY with scaling of CY . Note that we only need pl ﬂips here because every extremal ray contracted in the process

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would have negative intersection with some component of \BL by the properties of C mentioned in Step 2. By the special termination of Theorem 2.9, after ﬁnitely many steps, Supp \BL does not intersect the extremal rays contracted by the LMMP, hence we end up with a model Y D on which KY " + BY " is nef/Z. Clearly, (Y D /Z, BY " ) is a nef model of (X/Z, B), but may not be a log minimal model because the inequality in (1) of Deﬁnition 2.1 may not be satisﬁed. Step 4. Let T = {t ∈ [0, 1] | (X/Z, B + tC) has a log minimal model}. Since 1 ∈ T , T = K ∅. Let t ∈ T ∩(0, 1] and let (Yt /Z, BYt +tCYt ) be any log minimal model of (X/Z, B + tC). Running the LMMP/Z on KYt + BYt with scaling of tCYt shows that there is a tD ∈ (0, t) such that [tD , t] ⊂ T because the inequality required in (1) of Deﬁnition 2.1 is an open condition. The LMMP terminates for the same reasons as in Step 3 and we note again that the log ﬂips required are all pl ﬂips. Step 5. Let τ = inf T . If τ ∈ T , then by Step 4, τ = 0 and so we are done by deriving a contradiction. Thus, we may assume that τ ∈ / T . In this case, there is a sequence t1 > t2 > · · · in T ∩ (τ, 1] such that limk→+∞ tk = τ . For each tk let (Ytk /Z, BYtk + tk CYtk ) be any log minimal model of (X/Z, B + tk C) which exists by the deﬁnition of T and from which we get a nef model (YtDk /Z, BYt" + τ CYt" ) for k k (X/Z, B + τ C) by running the LMMP/Z on KYtk + BYtk + τ CYtk with scaling of (tk −τ )CYtk . Let D ⊂ X be a prime divisor contracted/YtDk . If D is contracted/Ytk , then a(D, X, B + tk C) < a(D, Ytk , BYtk + tk CYtk ) ≤ a(D, Ytk , BYtk + τ CYtk ) ≤ a(D, YtDk , BYt" + τ CYt" ), k

k

but if D is not contracted/Ytk we have a(D, X, B + tk C) = a(D, Ytk , BYtk + tk CYtk ) ≤ a(D, Ytk , BYtk + τ CYtk ) < a(D, YtDk , BYt" + τ CYt" ), k

k

because (Ytk /Z, BYtk + tk CYtk ) is a log minimal model of (X/Z, B + tk C) and (YtDk /Z, BYt" + τ CYt" ) is a log minimal model of (Ytk /Z, BYtk + τ CYtk ). Thus, in k k any case we have a(D, X, B + tk C) < a(D, YtDk , BYt" + τ CYt" ). k

k

Replacing the sequence {tk }k∈N with a subsequence, we can assume that all the induced rational maps X ''( YtDk contract the same components of B + τ C. Now an easy application of the negativity lemma (cf. [2], Claim 3.5) implies that the log discrepancy a(D, YtDk , BYt" + τ CYt" ) is independent of k. Therefore, each k k (YtDk , BYt" + τ CYt" ) is a nef model of (X/Z, B + τ C) such that k

k

a(D, X, B + τ C) = lim a(D, X, B + tl C) ≤ a(D, YtDk , BYt" + τ CYt" ) l→+∞

k

k

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for any prime divisor D ⊂ X contracted/YtDk . Step 6. To get a log minimal model of (X/Z, B + τ C) we just need to extract those prime divisors D on X contracted/YtDk for which a(D, X, B + τ C) = a(D, YtDk , BYt" + τ CYt" ). k

k

Since B has a component which is ample/Z, we can ﬁnd Δ on X such that Δ ∼R B + τ C/Z and such that (X/Z, Δ) and (YtDk /Z, ΔYt" ) are klt (see Remark 2.4). k Now we can apply Lemma 2.11 to construct a crepant model of (YtDk /Z, ΔYt" ) k which would be a log minimal model of (X/Z, Δ). This in turn induces a log minimal model of (X/Z, B + τ C). Thus, τ ∈ T and this gives a contradiction. Therefore, W = ∅.

3. Nonvanishing In this section we are going to prove Theorem 1.5, which is a numerical version of the corresponding result obtained in [3] (see equally [18], [8] for interesting presentations of [3]). Preliminaries. During the following subsections, we will give a complete proof of the next particular case of Theorem 1.5 (the absolute case Z = {z}). Theorem 3.1. Let X be a smooth projective manifold, and let B be an R-divisor such that: (1) The pair (X, B) is klt, and B is big. (2) The adjoint bundle KX + B is pseudo-eﬀective. %N Then there exists an eﬀective R-divisor j=1 ν j [Yj ] numerically equivalent with KX + B. We recall that by deﬁnition a big divisor B contains in its cohomology class a current (1)

ΘB := ωB + [E],

where ωB is a K¨ ahler metric, and [E] is the current of integration associated to an eﬀective R-divisor E. This is just a reformulation of the usual Kodaira lemma, except that in algebraic geometry one usually denotes the decomposition (1) by B = H + E, where H is ample; the ωB above is a smooth, positive representative of c1 (H). Moreover, the pair (X, B) is klt and X is assumed to be non-singular, thus we have M (2) B= bj Zj ,

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where the bj are positive reals, (Zj ) is a ﬁnite set of hypersurfaces of X such that (3)

1

j

|fj |−2b ∈ L1 (Ω)

j

for each coordinate set Ω ⊂ X, where Zj ∩ Ω = (fj = 0). Therefore, by considering a convex combination of the objects in (1) and (2), we can assume from the beginning that the R-divisor E satisfy the integrability condition (3): this can be seen as a metric counterpart of hypothesis (1) in statement 3.1. Let L be a pseudo-eﬀective R-divisor on X; we denote its numerical dimension by num(L). The formal deﬁnition will not be reproduced here (the interested reader can proﬁtably consult the references [23], [4]), however, in order to gain some intuition about it, let us mention that if L has a Zariski decomposition, then num(L) is the familiar numerical dimension of the nef part. The statements which will follow assert the existence of geometric objects in the Chern class of L and its approximations, according to the size of its numerical dimension. The ﬁrst one is due to N. Nakayama in [23] (see also the transcendental generalization by S. Boucksom, [4]). Theorem 3.2. ([4], [23]) Let L be a pseudo-eﬀective R-divisor such that num(L) = 0. Then there exists an eﬀective R-divisor Θ :=

ρ M

ν j [Yj ] ∈ {α}.

j=1

For a more complete discussion about the properties of the divisor Θ above we refer to the article [23]. Concerning the pseudoeﬀective classes in NSR (X) whose numerical dimension is strictly greater than 0, we have the following well-known statement. Theorem 3.3. ([20]) Let X be a projective manifold, let L be a pseudo-eﬀective R-divisor, such that num(L) ≥ 1. Let B be a big R-divisor. Then for any x ∈ X and m ∈ Z+ there exist an integer km and a representative Tm,x := [Dm ] + ωm ≡ mL + B, where Dm is an eﬀective Q-divisor and ωm is a K¨ ahler metric such that ν(Dm , x) ≥ km and km → ∞ as m → ∞. Dichotomy. We start now the actual proof of 3.1 and denote by ν the numerical dimension of the divisor KX + B. We proceed as in [27], [16], [3], [34]. • If ν = 0, then Theorem 3.5 is a immediate consequence of 3.6, so this ﬁrst case is completely settled.

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• The second case ν ≥ 1 is much more involved; we are going to use induction on the dimension of the manifold. Up to a certain point, our arguments are very similar to the classical approach of Shokurov (see [27]); perhaps the main diﬀerence is the use of the invariance of plurigenera extension techniques as a substitute for the Kawamata-Viehweg vanishing theorem in the classical case. Let G be an ample bundle on X, endowed with a smooth metric whose curvature form is denoted by ωG ; by hypothesis, the R-divisor KX + B is pseudo-eﬀective, thus for each positive ε, there exists an eﬀective R-divisor ΘKX +B,ε ≡ KX + B + εG. We denote by Wε the support of the divisor ΘKX +B,ε and we consider a point x0 ∈ X \ ∪ε Wε . Then statement 3.3 provides us with a current T = [Dm ] + ωm ≡ m(KX + B) + B such that ν(Dm , x0 ) ≥ 1 + dim(X). The integer m will be ﬁxed during the rest of the proof. The next step in the classical proof of Shokurov would be to consider the logcanonical threshold of T , in order to use an inductive argument. However, under the assumptions of 3.1 we cannot use exactly the same approach, since unlike in the nef context, the restriction of a pseudo-eﬀective class to an arbitrary hypersurface may not be pseudo-eﬀective. In order to avoid such an unpleasant surprise, we introduce now our substitute for the log canonical threshold (see [26] for an interpretation of the quantity below, and also [3] for similar considerations). ) → X be a common log resolution of the singular part of T and ΘB . Let µ0 : X By this we mean that µ0 is the composition of a sequence of blow-up maps with non-singular centers, such that we have (4)

µp0 (ΘB ) =

M j∈J

(5)

µp0 (T ) =

M j∈J

(6)

KX/X = e

)B , ajB [Yj ] + Λ

)T , ajT [Yj ] + Λ

M j∈J

ajX/X [Yj ], e

where the divisors above are assumed to be non-singular and to have normal cross)B, Λ ) T are smooth (1,1)-forms. ings, and Λ Now the family of divisors ΘKX +B,ε enters into the picture. Let us consider its inverse image via the map µ0 : 7 G M j ) K +B,ε , aKX +B,ε [Yj ] + Λ µp0 ΘKX +B,ε = X j∈J

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) K +B,ε in the relation above is an eﬀective R-divisor, whose support does where Λ X not contain any of the hypersurfaces (Yj )j∈J . The set J is ﬁnite and given independently of ε, so we can assume that the following limit exists: ajKX +B := lim ajKX +B,ε . ε→0

For each j ∈ J, let αj be a non-singular representative of the Chern class of the bundle associated to Yj ; by the preceding equality we have M 7 G M j ) K +B,ε + δεj αj , aKX +B [Yj ] + Λ µp0 ΘKX +B,ε ≡ X

(7)

j∈J

j∈J

) the ε-free part of the where δεj := ajKX +B,ε − ajKX +B . Below, we denote by D current above. In conclusion, we have organized the previous terms such that µ0 appears as a partial log-resolution for the family of divisors (ΘKX +B,ε )ε>0 . Given any real number t, consider the following quantity: 7 G µp0 KX + t(T − ΘB ) + ΘB ; it is numerically equivalent to the current ) + (1 − t)Λ ) B + tΛ )T + KXe + (1 + mt)D

M

γ j (t)[Yj ],

j∈J

where we use the following notation: γ j (t) := tajT + (1 − t)ajB − (1 + mt)ajKX +B − ajX/X . e G 7 j ) +Λ ) K +B,ε + % We have µp ΘKX +B ≡ D X j∈J δε αj , and on the other hand the cohomology class of the current G 7 t(T − ΘB ) + ΘB − (1 + mt) ΘKX +B,ε − εωG is equal to the ﬁrst Chern class of X, so by the previous relations we infer that the currents M G 7 ) K +B,ε + (8) (1 + mt) Λ ωG δεj αj − ε) X j∈J

and (9)

Θωb (KXe ) +

M

) B + tΛ )T γ j (t)[Yj ] + (1 − t)Λ

j∈J

are numerically equivalent, for any t ∈ R. We use next the strict positivity of B, in order to modify slightly the inverse image of ΘB within the same cohomology class, so that we have:

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Minimal models, ﬂips and ﬁnite generation

(i) The real numbers

− ajB 1 + ajKX +B + ajX/X b ajT − ajB − majKX +B

are distinct. < 1. (ii) The klt hypothesis in 3.1 is preserved, i.e., ajB − ajX/X e ) B } contains a K¨ (iii) The (1,1)-class {Λ ahler metric. The arguments we use in order to obtain the above properties are quite standard: it is the so-called tie-break method. Nothing very sophisticated is needed here; however, one can proﬁtably consult, e.g., the articles of O. Fujino and F. Ambro in [13] for more on this technique. Granted this, there exists a unique index, say 0 ∈ J and a positive real τ such that γ 0 (τ ) = 1 and γ j (τ ) < 1 for j ∈ J \ {0}. Moreover, we have 0 < τ < 1, by the klt hypothesis and the concentration of the singularity of T at the point x0 . We equally have the next numerical identity M 7 G ) ≡ K e + S) + B, ) ) K +B,ε + (10) (1 + mτ ) Λ δεj αj − εωG + H X X j∈J

) is the R-divisor where B (11)

) := B

M

)B + τ Λ )T γ j (τ )[Yj ] + (1 − τ )Λ

j∈Jp

and we also denote (12)

) := − H

M

γ j (τ )[Yj ].

j∈Jn

The choice of the partition of J = Jp ∪ Jn ∪ {0} is such that the coeﬃcients of ) is eﬀective, and of course the the divisor part in (11) are in [0, 1[, the R divisor H 0 ) coeﬃcient γ (τ ) = 1 corresponds to S. In order to apply induction, we collect here the main features of the objects constructed above. % • In the ﬁrst place, the R-divisor j∈Jp γ j (τ )[Yj ] is klt, and the smooth (1, 1))B + τ Λ ) T is positive deﬁnite; thus the R-divisor in (11) is big and klt. form (1 − τ )Λ Moreover, its restriction to S) has the same properties. • There exists an eﬀective R-divisor Δ such that 7 G ) + S) + Δ ≡ µp0 B + τ m(KX + B) . B

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C. Birkar, M. P˘ aun

Indeed, the expression of Δ is easily obtained as follows M 7 M G [Yj ] + (1 + mτ )ajKX +B + ajX/X Δ := (τ ajT + (1 − τ )ajB )[Yj ]. e j∈Jn

j∈Jp ∪{0}

Therefore, it is enough to produce an eﬀective R-divisor numerically equivalent to ) in order to complete the proof of Theorem 3.1. KXe + S) + B ) and its restriction to S) are pseudo-eﬀective by • The adjoint bundle KXe + S) + B the relation (10). • By using a sequence of blow-up maps, we can even assume that the components (Yj )j∈Jp in the decomposition (11) have empty mutual intersections. Indeed, this is a simple—but nevertheless crucial!—classical result, which we recall next. ) such We denote by Ξ an eﬀective R-divisor, whose support does not contain S, ) that Supp Ξ ∪ S has normal crossings and such that its coeﬃcients are strictly smaller than 1. # →X ) such Lemma 3.4. (See, e.g., [13].) There exists a birational map µ1 : X that µp1 (KXe + S) + Ξ) + EXb = KXb + S + Γ, where EXb and Γ are eﬀective with no common components, EXb is exceptional, and ) moreover, the support of the divisor Γ has normal S is the proper transform of S; crossings, its coeﬃcients are strictly smaller than 1, and the intersection of any two components is empty. % We apply this result in our setting with Ξ := j∈Jp γ j (τ )[Yj ], and we summarize the discussion of this paragraph in the next statement (in which we equally adjust the notations). # → X and an R-divisor Proposition 3.5. There exist a birational map µ : X M #≡ #B B ν j Yj + Λ j∈J

# where 0 < ν j < 1, the hypersurfaces Yj above are smooth, they have empty on X, mutual intersection and moreover the following hold: (1) There exists a family of closed (1, 1)-currents Θε := ΔKX +B,ε + αε # where S ⊂ X # is a non-singular hynumerically equivalent with KXb + S + B, persurface which has transversal intersections with (Yj ), and where ΔKX +B,ε is an eﬀective R-divisor whose support is disjoint from the set (S, Yj ), and ﬁnally αε is a non-singular (1,1)-form, greater than −εω. # →X ) such that S is not µ1 -exceptional, and such (2) There exists a map µ1 : X # ) via µ1 . that ΛB is greater than the inverse image of a K¨ ahler metric on X # # Therefore, the form ΛB is positive deﬁnite at the generic point of X, and so is its restriction to the generic point of S.

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Minimal models, ﬂips and ﬁnite generation

# such that (3) There exists an eﬀective R-divisor Δ on X 7 G # + S + Δ = µp B + τ m(KX + B) + E, B where E is µ-exceptional. Restriction and induction. We consider next the restriction to S of the currents Θε above: #|S ; Θε|S ≡ KS + B

(13)

we have the following decomposition: M ρε,j Yj|S + Rε , (14) Θε|S = j∈J

where the coeﬃcients (ρε,j ) are positive real numbers, and Rε above is the closed current given by the restriction to S of the diﬀerential form (15)

αε

plus the part of the restriction to S of the R-divisor (16)

ΔKX +B,ε

which is disjoint from the family (Yj|S ). Even if the diﬀerential form in (15) may not be positive, nevertheless we can assume that we have Rε ≥ −εω|S for any ε > 0. We remark that the coeﬃcients ρε,j in (14) may be positive, despite of the fact the Yj does not belong to the support of Θε , for any j ∈ J. For each index j ∈ J we will assume that the following limit ρ∞,j := lim ρε,j ε→0

exists, and we introduce the following notation: I := {j ∈ J : ρ∞,j ≥ ν j }.

(17)

The numerical identity in 3.5, (1) restricted to S coupled with (14) shows that we have M M (ρε,j − ρ∞,j )[Yj|S ] ≡ KS + BS , (ρ∞,j − ν j )[Yj|S ] + Rε + (18) j∈J

j∈I

where BS is the current

M j∈J\I

# B|S . (ν j − ρ∞,j )[Yj|S ] + Λ

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C. Birkar, M. P˘ aun

We are now in a good position to apply induction: • The R-divisor BS is big and klt on S. Indeed, this follows by (2) and the # in 3.5, and the deﬁnition of the set I, see (17). properties of B • The adjoint divisor KS + BS is pseudoeﬀective, by passing to the limit in the relation (18). Therefore, we can apply the induction hypothesis: there exists a non-zero, eﬀective R-divisor, which can be written as M λi [Wi ] TS := i∈K

(where Wi ⊂ S are hypersurfaces), which is numerically equivalent to KS + BS . We consider now the current M M M ν j [Yj|S ]; λi [Wi ] + ρ∞,j [Yj|S ] + (19) T#S := i∈K

j∈J\I

j∈I

from the relation (18) we get #|S . T#S ≡ KXb + S + B

(20)

It is precisely the R-divisor T#S above which will “generate” the section we seek, in the following manner. We ﬁrst use a diophantine argument, in order to obtain # with a Q-divisor, respectively Q-line a simultaneous approximation of TS and B bundle, such that the relation (20) above still holds. The next step is to use a trick by Shokurov (adapted to our setting) and ﬁnally the main ingredient is an extension result for pluricanonical forms. All this will be presented in full details in the next three subsections. Approximation. In this paragraph we recall the following diophantine approximation lemma (we refer to [26] for a complete proof). #η Lemma 3.6. For each η > 0, there exist a positive integer qη , a Q-line bundle B # on X and a Q-divisor M M M (21) T#S,η := νηj [Yj|S ] ρ∞,j λiη [Wi ] + η [Yj|S ] + i∈K

j∈J\I

j∈I

on S such that: #η is a genuine line bundle, and the numbers A.1 The multiple qη B (qη λiη )i∈K , are integers.

(qη νηj )j∈J ,

(qη ρ∞,j η )j∈J

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Minimal models, ﬂips and ﬁnite generation

#η|S . A.2 We have T#S,η ≡ KXb + S + B 7 G 7 G #−B #η p < η, |qη λiη − λi | < η and the analogous relation for A.3 We have pqη B the (ρ∞,j , ν j )j∈J (here p · p denotes any norm on the real N´eron-Severi space # of X). A.4 For each η0 > 0, there exists a ﬁnite family (ηj ) such that the class {KXb + #η }, where 0 < ηj < η0 . # belongs to the convex hull of {K b + S + B S + B} j X Remark. Even if we do not reproduce here the arguments of the proof (again, see [26]), we present an interpretation of it, due to S. Boucksom. Let N := |J| + |K|; we consider the map l1 : RN → NSR (S) deﬁned as follows: to each vector (x1 , . . . , xN ), it associates the class of the R%|K| %N divisor i=1 xi Wi + j=1+|K| xj Yj|S . We deﬁne another linear map l2 : NSR (X) → NSR (S) which is given by the restriction to S. We are interested in the set I := (x1 , . . . , xN ; τ ) ∈ RN × NSR (X) such that l1 (x) = l2 (τ ); it is a vector space, which is moreover deﬁned over Q (since this is the case for both maps l1 and l2 ). Now our initial data (TS , {KX + S} + θLb ) corresponds to a point of the above ﬁbered product, and the claim of the lemma is that given a point in a vector subspace deﬁned over Q, we can approximate it with rational points satisfying the Dirichlet condition. A trick by V. Shokurov. Our concern in this paragraph will be to “convert” the eﬀective Q-divisor T#S,η 7 G #η . To this end, we will into a genuine section sη of the bundle qη KXb + S + B apply a classical argument of Shokurov, in the version revisited by Siu in his recent work [34]. A crucial point is that by a careful choice of the metrics we use, the L2 estimates will allow us to have very precise information concerning the vanishing of sη . Proposition 3.7. There exists a section Q 7 G3 #η|S sη ∈ H 0 S, qη KS + B whose zero set contains the divisor Q M 3 M j ν [Y ] qη ρ∞,j [Y ] + j|S j|S η η j∈J\I

for all 0 < η > 1.

j∈I

98 Proof of 3.7.

C. Birkar, M. P˘ aun

We ﬁrst remark that we have 7 G 7 G #η|S = KS + (qη − 1) KS + B #η|S + B #η|S ; q η KS + B

in order to use the classical vanishing theorems, we have to endow the bundle 7 G #η|S + B #η|S (qη − 1) KS + B with an appropriate metric. #η ; we will construct a metric on it induced by We ﬁrst consider the Q-bundle B the decomposition 7 G #η = B #+ B #η − B # . B The second term above admits a smooth representative whose local weights are η in C ∞ norm, by the approximation relation A.3. As for the ﬁrst bounded by qη one, we recall that we have M #B , #= ν j Yj + Λ (26) B j∈J

# B has the positivity properties as in Proposition 3.5, 2. where the (1,1)-form Λ #η|S is deﬁned such that its curvature Now, the ﬁrst metric we consider on B current is equal to M M G 7 # B|S + Ξ(η)|S , ν j Yj|S + Λ max ν j , νηj Yj|S + (27) j∈I

j∈J\I

# in the class of the current where Ξ(η) is a non-singular (1, 1)-form on X Q M G3 7 ν (j) − max ν (j) , νη(j) [Yj ] j∈I

#η − B; # we can assume that Ξ(η) is greater than −C η , where the constant plus B qη C above is independent of η. # B is semi-positive on X # and strictly positive at the generic The smooth term Λ point of S: thanks to these positivity properties we can ﬁnd a representative of the # B } which dominates a K¨ class {Λ ahler metric. In general we cannot avoid that this representative acquires some singularities. However, in the present context we will show that there exists a current in the above class which is “restrictable” to S. Indeed, we consider the exceptional divisors (Ej ) of the map µ1 (see Proposition 3.5); the hypersurface S does not belong to this set, and then the class M #B − Λ ε j Ej j

# for some positive reals εj . Once a set of such parameters is chosen, is ample on X, we ﬁx a K¨ ahler form M #B − ε j Ej } Ω ∈ {Λ j

99

Minimal models, ﬂips and ﬁnite generation

and for each δ ∈ [0, 1] we deﬁne (28)

M G 7 # B,δ := (1 − δ)Λ #B + δ Ω + ε j Ej . Λ j

For each η > 0, there exists a δ > 0 such that the diﬀerential form δΩ + Ξ(η) is positive deﬁnite. For example, we can take (29)

δ := C

η , qη

where the constant C > 0 does not depend on η. With the choice of several parameters as indicated above, the current # B,δ + Ξ(η) Λ # B,δ is in the same cohomology class as Λ #B, dominates a K¨ ahler metric, and since Λ we have M M 7 G #≡ # B,δ + Ξ(η). B max ν j , νηj Yj + ν j Yj + Λ (30) j∈I

j∈J\I

We remark that the current in the expression above admits a well-deﬁned restric# B,δ ) tion to S; moreover, the additional singularities of the restriction (induced by Λ η are of order C , thus it will clearly be klt as soon as η > 1. The current in the qη #η|S . expression (30) induces a metric with strictly positive curvature on B 7 G #η|S whose Next, we deﬁne a singular metric on the bundle (qη − 1) KS + B curvature form is equal to (qη − 1)T#S,η and we denote by hη the resulting metric on the bundle 7 G #η|S + Bη|S . (qη − 1) KS + B The divisor qη T#S,η corresponds to the current of integration along the zero set of the section uη of the bundle 7 G #η|S + ρ q η KS + B where ρ is a topologically trivial line bundle on S. By the Kawamata-Viehweg-Nadel vanishing theorem (cf. [15], [39], [22]) we have 7 G 7 G3 #η ⊗ I hη = 0 H j (S, qη KS + B 7 G #η + ρ, since ρ carries for all j ≥ 1, and the same is true for the bundle qη KS + B a metric with zero curvature. Moreover, the section uη belongs to the multiplier ideal of the metric hη above, as soon as η is small enough, because the multiplier #η|S will be trivial. Since the Euler characteristic ideal of the metric on the bundle B of the two bundles is the same, we infer that Q 7 G 7 G3 #η ⊗ I hη K= 0. H 0 S, qη KS + B

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C. Birkar, M. P˘ aun

We denote by sη any non-zero element in the group above; we show now that its zero set satisﬁes the requirements in the proposition. Indeed, locally at any point of x ∈ S we have A |fs |2 dλ < ∞, F (j) F ∞,j 2νηj (qη −1)+2e νηj 2ρη (qη −1)+2e νη (S,x) j∈I |fj | j∈J\I |fj | where ν)ηj := ν j if j ∈ J \ I and ν)ηj := max{νηj , ν j } if j ∈ I; we denote by fs the local expression of the section sη , and we denote by fj the local equation of Yj ∩ S. But then we have A |fs |2 dλ < ∞ F F ∞,j 2νηj qη 2ρη qη (S,x) j∈I |fj | j∈J\I |fj | for all η > 1 (by the deﬁnition of the set I and the construction of the metric on #η|S ). Therefore, the proposition is proved. B # B,δ , we recall Remark 3.8. Concerning the construction and the properties of Λ the very nice result in [9], stating that if D is an R-divisor which is nef and big, then its associated augmented base locus can be determined numerically. Remark 3.9. As one can easily see, the divisor we are interested in in the previous proposition 3.7 is given by M M νηj [Yj|S ]. Eη := ρ∞,j η [Yj|S ] + j∈J\I

j∈I

The crucial fact about it is that it is smaller than the singularities of the metric #η ; this is the reason why we can infer that the section sη above we construct for B vanishes on qη Eη —and not just on the round down of the divisor (qη − 1)Eη —, see [26], page 42 for some comments about this issue. The method of Siu. We have arrived at the last step in our proof: for all 0 < η > 1, the section # Once this is done, we just use point A.4 of the sη admits an extension to X. approximation lemma 3.8, in order to infer the existence of an R-section of the # and then the relation (3) of 3.5 to conclude. bundle KXb + S + B, The extension of the section sη will be obtained by using the invariance of plurigenera techniques, thus in the ﬁrst paragraph of the current subsection, we #η constructed above. will highlight some of the properties of the Q-divisors B # η )η>0 . Uniformity properties of (KXb + S + B We list below the pertinent facts which will ultimately enable us to perform the extension of (sη ); the constant C which appears in the next statement is independent of η.

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Minimal models, ﬂips and ﬁnite generation

7 G #η ) vanishes along the divisor (U1 ) The section sη ∈ H 0 S, qη (KS + B Q M 3 M νηj [Yj|S ] ρ∞,j qη η [Yj|S ] + j∈I

j∈J\I

for all 0 < η > 1. #η } such that: (U2 ) There exists a closed (1,1)-current Θη ∈ {KXb + S + B η ω. qη (2.2) Its restriction to S is well deﬁned, and we have M θηj [Yj|S ] + Rη,S . Θη|S =

(2.1) It is greater than −C

j∈J

Moreover, the support of the divisor part of Rη,S is disjoint from the set (Yj|S ), and θηj ≤ ρ∞,j + C qηη . η #η can be endowed with a metric whose curvature current is (U3 ) The bundle B given by M # B,η + Ξ(η), νηj [Yj ] + Λ j∈J

where the hypersurfaces Yj above verify Yj ∩ Yi = ∅, if i K= j, and moreover we have: # B,η + Ξ(η) dominates a K¨ (3.1) The current Λ ahler metric. # B,η + Ξ(η) is well deﬁned, and if we denote by νη the (3.2) The restriction Λ |S maximal multiplicity of the above restriction, then we have qη νη ≤ Cη. Property (U1 ) is a simple recapitulation of facts which were completely proved during the previous paragraphs. The family of currents in (U2 ) can be easily obtained thanks to Proposition 3.5, by the deﬁnition of the quantities ρ∞,j and their approximations. #η as above is done precisely as in the Finally, the construction of the metric on B previous paragraph, except that instead of taking the coeﬃcients max(ν j , νηj ), we simply consider νηj . The negativity of the error term is the same (i.e., Cη/qη ). Let us introduce the following notations: % • Δ1 := j∈J\I νηj [Yj ]. It is an eﬀective and klt Q-bundle; notice that the multiple qη νηj is a positive integer strictly smaller than qη , for each j ∈ J \ I. % # B,η + Ξ(η). It is equally an eﬀective and klt Q-bundle • Δ2 := j∈I νηj [Yj ] + Λ such that qη Δ2 is integral.

102

C. Birkar, M. P˘ aun

Precisely as in [7], [10], [1], there exists a decomposition qη Δ1 = L1 + · · · + Lqη −1 such that for each m = 1, . . . , qη − 1, we have M Lm := Yj . j∈Im ⊂J\I

We denote Lqη := qη Δ2 and L(p) := p(KX + S) + L1 + · · · + Lp

(31)

where p = 1, . . . , qη . By convention, L(0) is the trivial bundle. We remark that it is possible to ﬁnd an ample bundle (A, hA ) independent of η whose curvature form is positive enough such that the next relations hold. (†) For each 0 ≤ p ≤ qη − 1, the bundle L(p) + qη A is generated by its global (p) sections, which we denote by (sj ). # (†2 ) Any section of the bundle L(qη ) + qη A|S admits an extension to X. (†3 ) We endow the bundle corresponding to (Yj )j∈J with a non-singular metric, and we denote by ϕ )m the induced metric on Lm . Then for each m = 1, . . . , qη , the functions ϕ )Lm + 1/3ϕA are strictly psh. 4

(† ) For any η > 0 we have (32)

Θη ≥ −

η ΘA . qη

Under the numerous assumptions/normalizations above, we formulate the next statement. Claim 3.10. There exists a constant C > 0 independent of η such that the section 7 G (p) 0 (p) s⊗k + kL(qη ) + qη A|S η ⊗ sj ∈ H S, L # for each p = 0, . . . , qη − 1, j = 1, . . . , Np and k ∈ Z+ such that extends to X, η k ≤ C. qη The statement above can be seen as a natural generalization of the usual invariance of plurigenera setting (see [5], [7], [10], [17], [19], [25], [33], [36], [38]); in substance, we are about to say that the more general hypothesis we are forced to consider induces an eﬀective limitation of the number of iterations we are allowed to perform.

Minimal models, ﬂips and ﬁnite generation

103

Proof of Claim 3.10. To start with, we recall the following very useful integrability criterion (see, e.g., [1]). Lemma 3.11. Let D be an eﬀective R-divisor on a manifold S. We consider the non-singular hypersurfaces Yj ⊂ S for j = 1, . . . , N such that Yj ∩ Yi = ∅ if i K= j, and such that the support of D is disjoint from the set (Yj ). Then there exists a constant ε0 := ε0 ({D}) depending only on the cohomology class of the divisor D such that for all positive real numbers δ ∈]0, 1] and ε ≤ ε0 we have A dλ F <∞ 2(1−δ) 2ε |f | D (S,s) j |fj | for all s ∈ S. In the statement above, we denote by fj , fD the local equations of Yj , respectively D near s ∈ S (with the usual abuse of notation). We will equally need the following version of the Ohsawa-Takegoshi theorem (see [6], [21], [24], [33]); it will be our main technical tool in the proof of the claim. # be a projective n-dimensional manifold, and let S ⊂ X # be Theorem 3.12. Let X a non-singular hypersurface. Let F be a line bundle, equipped with a metric hF . We assume that: √ −1 # (a) The curvature current ΘF is greater than a K¨ ahler metric on X. 2π (b) The restriction of the metric hF on S is well deﬁned. 7 G Then every section u ∈ H 0 S, (KXb + S + F|S ) ⊗ I(hF |S ) admits an extension U # to X. We will use inductively the extension theorem 3.12, in order to derive a lower bound for the power k we can aﬀord in the invariance of plurigenera algorithm, under the conditions (Uj )1≤j≤3 ; the ﬁrst steps are as follows. 7 G (0) Step 1. For each j = 1, . . . , N0 , the section sη ⊗ sj ∈ H 0 S, L(qη ) + qη A|S 7 G (q ) admits an extension Uj η ∈ H 0 X, L(qη ) + qη A , by the property †2 . (q )

Step 2. We use the sections (Uj η ) to construct a metric ϕ(qη ) on the bundle L(qη ) + qη A. G 7 (1) Step 3. Let us consider the section sη ⊗ sj ∈ H 0 S, L(1) + L(qη ) + qη A|S . We remark that the bundle L(1) + L(qη ) + qη A = KXb + S + L1 + L(qη ) + qη A can be written as KXb + S + F , where F := L1 + L(qη ) + qη A,

104

C. Birkar, M. P˘ aun

thus we have to construct a metric on F which satisﬁes the curvature and integrability assumptions in the Ohsawa-Takegoshi-type theorem above. Let δ, ε be positive real numbers; we endow the bundle F with the metric given by (q )

)L1 + (1 − ε)ϕ(qη ) + εqη (ϕA + ϕΘη ), ϕδ,εη := (1 − δ)ϕL1 + δ ϕ

(33)

where the metric ϕ )L1 is smooth (no curvature requirements, cf. †3 ) and ϕL1 is the weight of the singular metric induced by the divisors (Yj )j∈I1 . We denote by ϕΘη the local weight of the current Θη ; it induces a metric on the corresponding #η , which is used above. Q-bundle KXb + S + B We remark that the curvature conditions in the extension theorem will be fulﬁlled if δ < εqη , provided that η > 1: by the relations (†3 ) and (†4 ) the negativity of the curvature induced by the term δ ϕ )L1 will be absorbed by A. (1) Next we claim that the sections sη ⊗sj are integrable with respect to the metric deﬁned in (33), provided that the parameters ε, δ are chosen in an appropriate manner. Indeed, we have to prove that A S

(1)

G 7 |sη ⊗ sj |2 exp − (1 − δ)ϕL1 − εqη ϕΘη dV < ∞; % (0) 2 1−ε ( r |sη ⊗ sr | ) (0)

since the sections (sr ) have no common zeroes, it is enough to show that A G 7 |sη |2ε exp − (1 − δ)ϕL1 − εqη ϕΘη dV < ∞ S

(we have abusively removed the smooth weights in the above expressions, to simplify the writing). Now property (U1 ) concerning the zero set of sη is used: the above integral is convergent, provided that we have A M M G 7 νηj ϕYj ) dV < ∞. exp − (1 − δ)ϕL1 − εqη (ϕΘη − ρ∞,j η ϕYj − S

j∈I

j∈J\I

In order to check the convergence of the above integral, we would like to apply the integrability lemma 3.11; therefore, we have to estimate the coeﬃcients of the common part of the support of L1|S and M M νηj [Yj ] (34) Θη − ρ∞,j η [Yj ] − j∈I

j∈J\I

restricted to S. For any j ∈ J \ I, the coeﬃcient associated to the divisor Yj|S in the expression above is equal to (35)

θηj − ρ∞,j η

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Minimal models, ﬂips and ﬁnite generation

η . The singular part qη corresponding to j ∈ J \ I in expression (34) will be incorporated into (1 − δ)ϕL1 , thus we have to impose the relation and by property U2 , the diﬀerence above is smaller than C

1 − δ + qη εC

η < 1. qη

In conclusion, the positivity and integrability conditions will be satisﬁed provided that (36)

Cηε < δ < εqη ≤ ε0

We can clearly choose the parameters δ, ε such that (36) is veriﬁed. (q +1) Step 4. We apply the extension theorem and we get Uj η , whose restriction (1)

to S is precisely sη ⊗ sj . The claim will be obtained by iterating the procedure (1)-(4) several times, and estimating carefully the inﬂuence of the negativity of Θη on this process. Indeed, assume that we already have the set of global sections 7 G (kq +p) # L(p) + kL(qη ) + qη A Uj η ∈ H 0 X, (p)

which extend s⊗k η ⊗ sj . They induce a metric on the above bundle, denoted by (kqη +p) ϕ . If p < qη − 1, then we deﬁne the family of sections (p+1)

s⊗k η ⊗ sj

∈ H 0 (S, L(p+1) + kL(qη ) + qη A|S )

on S. As in step (3) above we remark that we have L(p+1) = KXb + S + Lp+1 + L(p) , thus according to the extension result 3.12, we have to exhibit a metric on the bundle F := Lp+1 + L(p) + kL(qη ) + qη A for which the curvature conditions are satisﬁed, and such that the family of sections above are L2 with respect to it. We deﬁne (37) G 7 1 (kq +p+1) ) (p) := (1 − δ)ϕLp+1 + δ ϕ )Lp+1 + (1 − ε)ϕ(kqη +p) + εqη kϕΘη + ϕA + ϕ ϕδ,ε η qη L and we check now the conditions that the parameters ε, δ have to satisfy. We have to absorb the negativity in the smooth curvature terms in (37) and the one from Θη . The Hessian of the term 1/3ϕA +

1 ϕ ) (p) qη L

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is assumed to be positive by †3 , but we also have a huge negative contribution −Ck

η ΘA qη

induced by the current Θη . However, we remark that we can assume that we have (38)

Ck

η < 1/3 qη

since this is precisely the range of k for which we want to establish the claim. Then the curvature of the metric deﬁned in (37) will be positive, provided that δ < εqη again by (†3 ). Let us check next the L2 condition; we have to show that the integral below is convergent: A S

(p+1)

G 7 |s⊗k ⊗ sj |2 exp − (1 − δ)ϕLp+1 − kqη εϕΘη dV. % ⊗k (p) 2 1−ε ( r |s ⊗ sr | )

This is equivalent with A 7 G |s|2εk exp − (1 − δ)ϕLp+1 − kqη εϕΘη dV < ∞. S

In order to show the above inequality, we use the same trick as before: the vanishing set of the section sη as in (U1 ) will allow us to apply the integrability lemma—the computations are strictly identical with those discussed in point 3) above, but we give here some details. By the vanishing properties of the section sη , the ﬁniteness of the previous integral will be implied by the inequality A M M G 7 νηj ϕYj ) dV < ∞. ρ∞,j exp − (1 − δ)ϕLp+1 − kεqη (ϕΘη − η ϕYj − S

j∈J\I

j∈I

In the ﬁrst place, we have to keep the poles of kεqη Θη “small” in the expression of the metric (37), thus we impose kεqη ≤ ε0 . The hypothesis in the integrability lemma will be satisﬁed provided that 1 − δ + εkqη C

η <1 qη

(this is the contribution of the common part of Supp Lp+1 and Θη ). Combined with the previous relations, the conditions for the parameters become (39)

Cεkη < δ < εqη < ε0 /k.

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Again we see that the inequalities above are compatible if k satisfy the inequality Ckη < qη which is precisely what Claim (3.10) states. In conclusion, we can choose the parameters ε, δ so that the integrability and positivity conditions in the extension theorem are veriﬁed; for example, we can take ε0 • ε := 2kqη and 7 η G ε0 • δ := 1 + kC . qη 4k Finally, let us indicate how to perform the induction step if p = qη − 1: we consider the family of sections (0)

⊗ sj sk+1 η

∈ H 0 (S, (k + 1)L(qη ) + qη A|S ),

In the case under consideration, we have to exhibit a metric on the bundle Lqη + L(qη −1) + kL(qη ) + qη A; however, this is easier than before, since we can simply take ϕqη (k+1) := qη ϕΔ2 + ϕ(kqη +qη −1) ,

(40)

where the metric on Δ2 is induced by its expression in the preceding subsection. With this choice, the curvature conditions are satisﬁed; as for the L2 ones, we remark that we have A A 7 G G 7 (0) 2 (0) qη (k+1) |sk+1 ⊗ s | exp − ϕ dV < C |sη ⊗ sj |2 exp − qη ϕΔ2 dV ; η j S

S

moreover, by the vanishing of sη along the divisor qη

7M

G νηj [Yj|S ] ,

j∈I

the right hand side term of the inequality above is dominated by A C

S

7 G exp − qη ϕΛ b B,η dV,

where the last integral is convergent because of the fact that qη ν < 1, see (U3 ). The proof of the extension claim is therefore ﬁnished.

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End of the proof. # as soon as η is small enough, We show next that the sections sη can be lifted to X by using Claim 3.10. (kq ) (0) Indeed, we consider the extensions Uj η of the sections s⊗k η ⊗ sj ; they can be used to deﬁne a metric on the bundle #η ) + qη A kqη (KXb + S + B (η)

whose kqηth root is deﬁned to be hk .

#η ), as an adjoint We write the bundle we are interested in, i.e., qη (KXb + S + B bundle, as follows: #η ) + B #η #η ) = K b + S + (qη − 1)(K b + S + B qη (KXb + S + B X X and this last expression equals 7 G #η + 1/kA + B #η − qη − 1 A. KXb + S + (qη − 1) KXb + S + B k Given the extension theorem 3.12, we need to construct a metric on the bundle 7 G #η + 1/kA + B #η − qη − 1 A. (qη − 1) KXb + S + B k (η)

On the ﬁrst factor of the above expression we will use (qη − 1)ϕk (that is to say, (η) the (qη − 1)th power of the metric given by hk ). #η with a metric whose curvature is given by the expresWe endow the bundle B sion M M # B,δ + Ξ(η); νηj [Yj ] + νηj [Yj ] + Λ j∈J\I

j∈I

#η|S here we take δ independent of η, but small enough such that the restriction B qη −1 is still klt. Finally, we multiply with k times h−1 A . H 9 By Claim 3.10, we are free to choose k, e.g., such that k = qη η −1/2 (where [x] denotes the integer part of the real x). Then the metric above is not identically ∞ when restricted to S, and its curvature will be strongly positive as soon as #η is greater than a K¨ # which is η > 1. Indeed, the curvature of B ahler metric on X # B,δ . independent of η because of the factor Λ 2 Moreover, the L conditions in Theorem 3.12 are satisﬁed, since the norm of (η) the section sη with respect to the metric qη ϕk is pointwise bounded, and by the #η|S . choice of the metric on B In conclusion, we obtain an extension of the section sη , and Theorem 1.5 is completely proved.

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The relative case. We will explain along the next lines the nonvanishing result 1.5 in its general form; to this end, we ﬁrst review the notion of relative bigness from the metric point of view. Let p : X → Z be a projective map and let B be an R-divisor on X. The pair (X, B) is klt by hypothesis, so we can assume that X is non-singular and that (41)

B=

N M

aj Wj ,

j=1

where 0 < aj < 1 and (Wj ) have normal crossings. Moreover, it is enough to prove 1.5 for non-singular manifolds Z (since we can desingularize it if necessary, and modify further X). The R-divisor B is equally p-big, thus there exist an ample bundle AX , an eﬀective divisor E on X and an ample divisor AZ on Z such that (42)

B + pp AZ ≡ AX + E.

By a suitable linear combination of the objects given by the relations (41) and (42) above, we see that there exists a klt current ΘB ∈ {B + pp AZ }, which is greater than a K¨ ahler metric. Thus, modulo the inverse image of a suitable bundle, the cohomology class of B has precisely the same metric properties as in the absolute case. The main technique we will use in order to settle 1.5 in full generality is the positivity properties of the twisted relative canonical bundles of projective surjections; more precisely, the result we need is the following. Theorem 3.13. ([1]) Let p : X → Z be a projective surjection, and let L → X be a line bundle endowed with a metric hL with the following properties. (1) The curvature current of (L, hL ) is positive. (2) There exist a generic point z ∈ Z, an integer m and a non-zero section u ∈ H 0 (Xz , mKXz + L) such that A 7 ϕL G 2 |u| m exp − dλ < ∞. m Xz Then the twisted relative bundle mKX/Z + L is pseudo-eﬀective, and it admits a positively curved metric hX/Z whose restriction to the generic ﬁber of p is less singular than the metric induced by the holomorphic sections that verify the L2/m condition in (2) above.

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Given this result, the end of the proof of 1.5 goes as follows. A point z ∈ Z will be called very generic if the restriction of ΘB to the ﬁber Xz dominates a K¨ahler metric and its singular part is klt, and moreover if the sections of all multiples of rational approximations of KX + B restricted to Xz do extend near z. We see that the set of very generic points of z is the complement of a countable union of Zariski closed algebraic sets; in particular, it is non-empty. Let z ∈ Z be a generic point. The adjoint R-bundle KXz + B|Xz is pseudoeﬀective, thus by the absolute case of 1.5 we obtain an eﬀective R-divisor Θ :=

N M

ν j Wj

j=1

within the cohomology class of KXz + B|Xz . By diophantine approximation we obtain a family of Q-bundles (Bη ) and a family of non-zero holomorphic sections 7 G uη ∈ H 0 Xz , qη (KXz + Bη|Xz ) induced by the rational approximations of Θ (see 3.6, 3.8 above). With these data, Theorem 3.17 provides the bundle qη (KX/Z + Bη ) with a positively curved metric hX/Z , together with a crucial quantitative information: the section uη is bounded with respect to it. The last step is yet another application of the Ohsawa-Takegoshi type theorem 3.12. Indeed, we consider the bundle qη (KX + Bη ) + pp A, where A → Z is a positive enough line bundle, such that A − (qη − 1)KZ is ample. We have the decomposition G 7 qη (KX + Bη ) + pp A = KX + (qη − 1)(KX/Z + Bη ) + Bη + pp A − (qη − 1)KZ and we have to construct a metric on the bundle

G 7 F := (qη − 1)(KX/Z + Bη ) + Bη + pp A − (qη − 1)KZ

with curvature conditions as in 3.12. The ﬁrst term in the sum above is endowed qη − 1 with the multiple of the metric hX/Z . The Q-bundle Bη is endowed with qη the metric given by ΘB plus a smooth term corresponding to the diﬀerence Bη −B. Finally, the last term has a non-singular metric with positive curvature, thanks to the choice of A; one can see that with this choice, the curvature assumptions in 3.12 are satisﬁed. The klt properties of B are inherited by Bη ; thus we have A A 3 Q q −1 η ϕX/Z − ϕBη dλ ≤ C exp(−ϕBη )dλ < ∞. |uη |2 exp − qη Xz Xz In conclusion, we can extend uη to the whole manifold X by 3.12. The convexity argument in Lemma 3.6 ends the proof of the nonvanishing.

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111

4. Proof of the main results Proof (of Theorem 1.1). We proceed by induction on d. Suppose that the theorem holds in dimension d − 1 and let (X/Z, B) be a klt pair of dimension d such that B is big/Z. By taking a log resolution we may assume that X is Q-factorial. First assume that KX + B is pseudo-eﬀective/Z. Then, by Theorem 1.5 in dimension d, KX + B is eﬀective/Z. Theorem 1.4 then implies that (X/Z, B) has a log minimal model. Now assume that KX + B is not pseudo-eﬀective/Z and let A be a general ample/Z Q-divisor such that KX + B + A is klt and nef/Z. Run the LMMP/Z on KX + B with scaling of A. By Theorem 2.8, we end up with a Mori ﬁbre space for (X/Z, B). Proof (of Corollary 1.2). Let (X/Z, B) be a klt pair of dimension d and f : X → Z D a (KX + B)-ﬂipping contraction/Z. By (1) of Theorem 1.1, there is a log minimal model (Y /Z D , BY ) of (X/Z D , B). By the base point free theorem, (Y /Z D , BY ) has a log canonical model which gives the ﬂip of f . Proof (of Corollary 1.3). If KX + B is not eﬀective/Z, then the corollary trivially holds. So, assume otherwise. By [12] there exist a klt pair (S/Z, BS ) of dimension ≤ dim X with big/Z Q-divisor BS , and p ∈ N such that locally over Z we have H 0 (mp(KX + B)) @ H 0 (mp(KS + BS )) for any m ∈ N. By Theorem 1.1, we may assume that KS + BS is nef/Z. The result then follows as KS + BS is semi-ample/Z by the base point free theorem.

References [1] B. Berndtsson, M. P˘ aun; A Bergman kernel proof of the Kawamata subadjunction theorem. arXiv:math/0804.3884. [2] C. Birkar; On existence of log minimal models. Compositio Math., to appear. Available at arXiv:0706.1792v3. [3] C. Birkar, P. Cascini, C. Hacon, J. Mc Kernan; Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), 405–468. [4] S. Boucksom; Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. Ecole Norm. Sup. (4) 37 (2004), no. 1, 45–76. [5] B. Claudon; Invariance for multiples of the twisted canonical bundle. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 1, 289–300. [6] J.-P. Demailly; On the Ohsawa-Takegoshi-Manivel extension theorem. Proceedings of the Conference in honour of the 85th birthday of Pierre Lelong, Paris, September 1997, Progress in Mathematics, Birkh¨ auser, 1999. [7] J.-P. Demailly; K¨ ahler manifolds and transcendental techniques in algebraic geometry. Plenary talk and Proceedings of the Internat. Congress of Math., Madrid (2006), 34pp., volume I.

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[8] S. Druel; Existence de mod`eles minimaux pour les vari´et´es de type g´en´eral. Expos´e 982, S´eminaire Bourbaki, 2007/08. [9] L. Ein, R. Lazarsfeld, M. Mustat¸˘ a, M. Nakamaye, M. Popa; Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701–1734. [10] L. Ein, M. Popa; Adjoint ideals and extension theorems. Preprint in preparation; june 2007. [11] O. Fujino; Special termination and reduction to pl ﬂips. In ”Flips for 3-folds and 4-folds” (ed. A. Corti), Oxford University Press (2007). [12] O. Fujino, S. Mori; A canonical bundle formula. J. Diﬀerential Geometry 56 (2000), 167–188. [13] C. Hacon, J. McKernan; Extension theorems and the existence of ﬂips. In ”Flips for 3-folds and 4-folds” (ed. A. Corti), Oxford University Press (2007). [14] G.H. Hardy, E.M. Wright; An introduction to the theory of numbers. Oxford University Press, 1938. [15] Y. Kawamata; A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann. 261 (1982), 43–46. [16] Y. Kawamata; Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79 (1985), no. 3. [17] Y. Kawamata; On the extension problem of pluricanonical forms. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 193–207. [18] Y. Kawamata; Finite generation of a canonical ring. arXiv:0804.3151. [19] D. Kim; Ph.D. Thesis. Princeton, 2006. [20] R. Lazarsfeld; Positivity in Algebraic Geometry. Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. [21] J. McNeal, D. Varolin; Analytic inversion of adjunction: L2 extension theorems with gain. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 3, 703–718. [22] A.M. Nadel; Multiplier ideal sheaves and K¨ ahler-Einstein metrics of positive scalar curvature. Ann. of Math. (2) 132 (1990), no. 3, 549–596. [23] N. Nakayama; Zariski decomposition and abundance. MSJ Memoirs 14, Tokyo (2004). [24] T. Ohsawa, K. Takegoshi; On the extension of L2 holomorphic functions. Math. Z., 195 (1987), 197–204. [25] M. P˘ aun; Siu’s Invariance of Plurigenera: a One-Tower Proof. Preprint IECN (2005), J. Diﬀerential Geom. 76 (2007), no. 3, 485–493. [26] M. P˘ aun; Relative critical exponents, non-vanishing and metrics with minimal singularities. arXiv:0807.3109. [27] V. Shokurov; A non-vanishing theorem. Izv. Akad. Nauk SSSR (49) 1985. [28] V.V. Shokurov; Three-dimensional log ﬂips. With an appendix in English by Yujiro Kawamata. Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. [29] V.V. Shokurov; 3-fold log models. Algebraic geometry, 4. J. Math. Sci. 81 (1996), no. 3, 2667–2699.

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[30] V.V. Shokurov; Prelimiting ﬂips. Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82–219; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 75–213. [31] V.V. Shokurov; Letters of a bi-rationalist VII. Ordered termination. arXiv:math/ 0607822v2. [32] Y-T. Siu; Invariance of plurigenera. Invent. Math. 134 (1998), no. 3, 661–673. [33] Y.-T. Siu; Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. Complex geometry (G¨ ottingen, 2000), 223–277, Springer, Berlin, 2002. [34] Y.-T. Siu; A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring. arXiv:math/0610740. [35] Y.-T. Siu; Finite Generation of Canonical Ring by Analytic Method. arXiv:0803.2454. [36] S. Takayama; Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165 (2005), no. 3, 551–587. [37] H. Tsuji; Extension of log pluricanonical forms from subvarieties. math.CV/0511342. [38] D. Varolin; A Takayama-type extension theorem. math.CV/0607323, to appear in Comp. Math. [39] E. Viehweg; Vanishing theorems. J. Reine Angew. Math. 335 (1982). Caucher Birkar, Cambridge, UK E-mail: [email protected] Mihai P˘ aun, Nancy, France E-mail: [email protected]

Remarks on an Example of K. Ueno Fr´ed´eric Campana 1. Introduction In [U75], Example 16.15, p. 207, Kenji Ueno constructs the threefold X described below, and says that it is unknown, whether or not it is unirational. After recalling Ueno’s construction, we show that X is rationally connected1 . The (much deeper) question of whether X is unirational (or even rational) remains open. We found it worthwhile drawing attention to this seemingly unnoticed example, on which several questions (such as unirationality vs rational connectedness), might possibly be tested more easily than on the test cases usually considered (such as quartics or double P3 ’s ramiﬁed along sextics). We formulate some of these questions in §4.

2. Ueno’s Example √ Let E := C/Z[i] be the elliptic curve with complex multiplication by i := −1. The order 4 cyclic group H generated by this multiplication has 4 points on E with non-trivial isotropy: the images of x = 0, (1 + i)/2, which are ﬁxed by all of H, and the images of x = 1/2, i/2, which have isotropy of order 2. Let A := E × E × E be the product of three copies of E, with three coordinates (x, y, z) in E. We deﬁne the automorphism θ of A by: θ(x, y, z) := (ix, iy, iz). Let G be the cyclic group of order 4 generated by θ, and X := A/G be the quotient of A by G. Let also g : A → X be the quotient map. Since G commutes with all automorphisms of A ﬁxing the origin (0, 0, 0) of A, the automorphism group of X contains a group isomorphic to SL(3, Z). The set of points of A having non-trivial isotropy under G consists of 43 = 64 points; out of these, 8 points, denoted eDj , have isotropy equal to G, and 56, denoted fkD , have isotropy of order 2, and these are exchanged in pairs by θ. The images by g of the ﬁrst 8 points are denoted with ej , they are singular points of type 41 (1, 1, 1) in M. Reid’s notation. These isolated singularities are logterminal, and non canonical. They are each resolved by one blow-up, which produces a smooth exceptional divisor isomorphic to P2 , with normal bundle O(−4). The images by g of the last 56 points are 28 points denoted with fk . They are singular points of type 21 (1, 1, 1). These isolated singularities are terminal, hence canonical. They are each resolved by one blow-up, which produces a smooth exceptional divisor isomorphic to P2 , with normal bundle O(−2). Altogether X has thus 8 + 28 = 36 singular points and g is ´etale outside of the complement of these in X. 1 In

1975, this notion of rational connectedness had not been introduced.

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Let hD : AD → A be the blow-up of A in the above 64 points eDj , fkD . Let be the corresponding exceptional divisors, all isomorphic to P2 . Then θ lifts uniquely to an automorphism, denoted θD , of AD . And GD , the group generated by θD , operates trivially on the EjD , and exchanges in pairs the divisors FkD , on which θD2 operates trivially. Let next g D : AD → X D := AD /GD be the quotient map, and h : X D → X the morphism such that hg D = ghD . Then h is nothing but the blow-up of X in its 36 singular points, and X D is smooth. Indeed, ‘linear’ coordinates (x, y, z) near a point eDj (resp. fkD ), give coordinates (x, v := y/x, w := z/x) on AD near a generic point of EjD (resp. FkD ). In such coordinates, θ acts by: θ(x, y, z) := (ix, iy, iz), and we get θD (x, v, w) := (ix, v, w) (resp. θD2 (x, v, w) = (−x, v, w)). We thus obtain local coordinates (u := x4 , v, w) (resp. (u := x2 , v, w)) on X D , near the corresponding point by g D . % % Lemma 2.1. We have KX " = −( 41 · j Ej ) + ( 12 · k Fk ).

EjD , FkD

Proof. The canonical bundle of X D is trivial on the complement of the EjD and FkD . The adjunction formula thus concludes the proof, since the EjD (resp. FkD ), are isomorphic to P2 , with normal bundles of degree −4 (resp. −2). Remark 2.2. 1. By contrast, KX is trivial. Since X D is rationally connected (as shown below), we have thus: κ(X D ) = −∞, and κ(X) = 0. In particular, X gives an example of a log-terminal threefold such that its Kodaira dimension is (strictly) larger than the Kodaira dimension of its resolutions (question raised by T. Peternell). Notice that if f : Y D → Y is a resolution of a projective normal variety Y , then κ(Y D ) ≤ κ(Y ), with equality if Y has only canonical singularities. 2. Ueno’s construction of course works in any dimension n. However, if n ≤ 2, X D is rational, while κ(X D ) = 0 if n ≥ 4. So only the case n = 3 is unclear for precise classiﬁcation purposes. 3. In §5, we shall consider the case when n = 2 in more detail. In this case, let S be the quotient (E × E)/G := S, where G is the cyclic group of order 4 generated by the automorphism θ of E × E deﬁned by: θ(x, y) = (ix, iy). Let also S D be the blow-up of S along its 10 singular points, so that S D is also the quotient of AD := E × E blown-up in its 16 points of order 2 by the lift of θ to AD .

3. Rational Connectedness Theorem 3.1. The threefold X D is rationally connected. Proof. This can be shown in many ways. For example, by establishing that H 0 (X D , SymN (ΩpX )) = 0, ∀N > 0, ∀p > 0. One can however easily avoid Miyaoka’s theorem that X D is uniruled if κ(X D ) = −∞, and the existence of minimal models in dimension 3. One could certainly construct explicitly rational curves joining two generic points of X D . However, even just constructing an explicit covering family of rational curves on X D is not obvious (even in dimension n = 2!). We shall thus proceed in a slightly more indirect way.

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The lemmas 3.2 and 3.3 below show that X D is uniruled and simply connected. Let us show how the result is implied by these two lemmas. Let r : X D → R be the ‘rational quotient’ of X D (called ‘MRC ﬁbration’ in Koll´ ar-Miyaoka-Mori’s terminology; it is characterised by having rationally connected ﬁbres and non-uniruled base by [GHS03]). We choose R smooth. Let d := dim(R). Since X D is uniruled, d ≤ 2. We shall show that d = 0. Since X D is simply connected, so is R. Assume that d = 1. Since the only simply connected curve is P1 , this case does not occur. Assume that d = 2. Since R is not uniruled, κ(R) ≥ 0 (by classiﬁcation). If κ(R) > 0, the line bundle L :=r∗ ((KR ))⊗m ⊂ Symm (Ω2X " ) has at least 2 linearly independent sections for m > 0 large and divisible. This is impossible, since every coherent subsheaf L of Symm (Ω2A ) has a space of sections bounded from above by its rank, since Symm (Ω2A ) is trivial. Thus κ(R) = 0. Since R is simply connected, R is birational to a K3 surface. Hence 1 = h2,0 (R) ≤ h2,0 (X D ). But this contradicts the fact that h2,0 (AD )G = 0. Thus d = 0, and X D is rationally connected. Lemma 3.2. The manifold X D is uniruled. Proof. Let X DD be the projective threefold obtained by blowing-up the points ej , but not the points fk on X. Let a be a generic point on X DD , there thus exists an irreducible curve C DD on X DD , complete intersection of generic ample divisors on X DD , and going through a, meeting the Ej ’s, but none of the points fk . Taking the strict transform C D on X D , we thus have: KX " · C D < 0, by the preceding lemma 2.1. From [Mi-Mo86], we get a rational curve through a, and X D is indeed uniruled. Lemma 3.3. The manifold X D is simply connected. Proof. It is suﬃcient to show that appropriate loops cDj , dDj , j = 1, 2, 3 on AD , and generating its fundamental group, have their image homotopically trivial in X D , and that the same property holds for some suitable path cD joining a and θ(a), for some arbitrary a in AD . Take for t ∈ [0, 1]: c1 (t) := (t, 0, 0) (resp. c2 (t) := (0, t, 0), resp. c3 (t) := (0, 0, t)), denoting also by t ∈ C its projection in E. We take for cj the continuous lift of cj to AD . Finally, deﬁne dDj := θ(cDj ). Notice that −cj = cj , hence the homotopic triviality of cj in the Kummer quotient A/m−1C, and so in X. The same conclusion thus holds for cDj , dDj in X D . For cD , just take any path joining any ﬁxed point of θD to itself, since its image in π1 (X D ) is in the (trivial) image of the subgroup generated by the cDj , dDj . Remark 3.4. Lemma 3.3 is only used to exclude R being birationally an Enriques surface. One could replace it by the vanishing of H 0 (X D , Sym2 (ΩX " )), which requires more computations.

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Remark 3.5. The generic irreducible rational curves C of covering families of X D go through at least one of the 8 non-canonical points of X (because KX .C D < 0, D D if C ⊂ X is its strict transform). Their inverse images in A are stable by θ, and thus admit complex multiplication by i.

4. Some questions We formulate a number of questions related to the example. 1. Determine a ‘Mori model’ of X D , that is, a normal terminal threefold Y birational to X D , obtained from X D by a succession of divisorial contractions and ‘ﬂips’, and ‘minimal’ in the sense that it has no birational Mori contraction. (Even in the surface case this is not entirely immediate. See §5 below for details: the Y obtained after blowing down twelve (−1)-curves is isomorphic to P1 × P1 ). 2. Decide whether Y is Q-Fano, a Del Pezzo ﬁbration, or a conic bundle. 3. Does X D admit non-trivial rational ﬁbrations with rationally connected ﬁbres? (It has countably many ﬁbrations in elliptic curves or Abelian surfaces, and is so ‘less birationally rigid’ than some threefolds exhibited by I. Cheltsov). See question 6 below. 4. Is X D rational, or unirational? The (non) rationality of X D could possibly be decided on one of its Mori Models. 5. The potential non-unirationality of X D might be more easily checked than on the usual examples (quartics in P4 or double covers of P3 ramiﬁed along a sextic). One possible approach might be the following variant of the negative solution of the L¨ uroth problem for the three-dimensional quartic due to Iskovskih-Manin. Observe indeed that any unirational variety Y (of dimension n ≥ 2) has an inﬁnite-dimensional space of rational dominant endomorphisms of a certain, ﬁxed, degree. For this, consider any dominant projection rational map p : Y ''( Pn of degree d, and any dominant unirationality map u : Pn ''( Y , of degree δ. For any birational automorphism f : Pn ''( Pn , consider now all the compositions u ◦ f ◦ p : Y ''( Y , for varying f , all of degree δ · d. In the special case of the Ueno threefold X, the existence of so many rational endomorphisms of ﬁxed degree should be compatible with the covering map g : A → X, ´etale outside a ﬁnite subset. The contrast to the two-dimensional case, where these endomorphisms do exist, lies in the rigidity of the three-dimensional singularities of X. 6. Let X be a rationally connected threefold. Does it have a non-trivial rational ﬁbration with generic ﬁbre Xy having either κ(Xy ) = 0, or κ(Xy ) = −∞?

5. The two-dimensional case When n = 2, consider the smooth surface S D := AD /G deﬁned in Remark 2.2(3), i.e., the two-dimensional version of Ueno’s threefold. We shall give here an explicit description of S D . Notice that S D contains a group of automorphisms naturally

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isomorphic to SL(2, Z) acting faithfully on its Picard Lattice (so with positive entropy). Notice that S D is a rational surface with Picard number 14, intermediate between the Picard numbers of the McMullen examples and of the Coble surface. Proposition 5.1. The surface S D is isomorphic to P1 ×P1 blown-up in twelve points (some inﬁnitesimally near) as follows. First blow up the 8 points with coordinates (s, t) equal to: (1, ±1), (−1, ±i), a := (∞, 0), aD := (0, 0), b := (∞, ∞), bD := (0, ∞). Denote by ¯bD , a ¯, a ¯D , ¯b the exceptional divisors of the resulting surface lying D above the 4 points b , a, aD , b respectively. Finally, blow up the resulting surface at the intersection of the divisors ¯bD , a ¯, a ¯D , ¯b with the strict transforms of the lines of equation s = 0, s = ∞, s = 0, s = ∞ respectively. Proof. We shall only sketch it. Notice ﬁrst that the surface S has ten singular points, 4 being of type 41 · (1, 1), and 6 of type 21 · (1, 1) in M. Reid’s notation. The 1+i 4 ﬁrst ones are the images of the points of coordinates (0, 0), (0, 1+i 2 ), ( 2 , 0) and 1+i 1+i D D ( 2 , 2 ), and are denoted A, B, B , A respectively (the double use of the letter A should not lead to confusion). The 6 last ones are respectively the images of 1+i i i i 1 i the points of coordinates (0, 2i ), ( 2i , 0), ( 2i , 1+i 2 ), ( 2 , 2 ), ( 2 , 2 ), ( 2 , 2 ), and are D D D respectively denoted a, b , b, a , c, c . (A drawing is useful here.) Blowing-up these 10 points, and denoting the 10 corresponding exceptional divisors on S D by the same letter as the points, we obtain 4 (−4)-curves and 6 (−2)-curves, and KS " = − 12 .(A + AD + B + B D ), from which we get: KS2 " = − 14 .[4 + 4 + 4 + 4] = −4. Since S D is a smooth rational surface, its Picard number is: 1 + (9 − (−4)) = 14. We shall now consider, in addition, the following 10 rational curves on S D . 1. The images D+ , D− , Di+ , Di− of the strict transforms in AD of the graphs in E × E of the multiplications by 1, −1, i, −i respectively. A simple computation shows that these 4 disjoint curves are (−1)-curves, and that moreover, D+ and D− (resp. Di+ and Di− ) meet transversally in one point each of the divisors A, AD , c (resp. A, AD , cD ). 2. The images V, V D , H, H D in S D of the strict transforms in AD of the four elliptic 1+i curves {0} × E, { 1+i 2 } × E, E × {0}, E × { 2 }. One checks again that these four disjoint curves are (−1)-curves on S D , disjoint from the four curves D above. Each of these 4 curves meets transversally in one point 3 of the exceptional divisors, which are respectively: (A, a, B); (B D , aD , AD ); (A, bD , B D ); and (B, b, AD ). 3. The images W, J of the strict transforms in AD of the two elliptic curves 1 { 2 } × E and E × { 21 } respectively. These are (−2)-curves meeting transversally in one point each 4 of the exceptional divisors, repectively: (a, c, cD , aD ) and (b, c, cD , bD ). We now ﬁrst blow down the 8 pairwise disjoint (−1)-curves D± , Di± , V, V D , D D H, H D . The image points will be denoted by: d± , d± i , v, v , h, h , respectively. The D D D D D image curves of A, A , B, B , a, a , b, b , c, c , W, J will be denoted by the very same symbols. One checks that their self-intersection numbers are respectively: 2, 2, −2, −2, −1, −1, −1, −1, 0, 0, −2, −2. This is obtained by directly computing the number of blown-down curves meeting each of these 12 curves, since all intersections are transversal. Notice that the (new) curves (A, a, B) (resp. (AD , b, B); (AD , aD , B D );

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(A, bD , B D )) meet transversally at v (resp. hD ; v D ; h). Similarly, the curves (A, AD , c) (resp. (A, AD , cD )) meet transversally at d± (resp. at d± i ). The (new) curves W (resp. J) meet transversally at one single point the curves (b, c, bD ) (resp. (a, cD , aD )), and no other. The 4 curves a, aD , b, bD , being pairwise disjoint (−1)-curves, can be blowndown simultaneously, leading to a smooth rational surface Y of Picard number 2. We denote the image points and curves by the very same symbols as before this second blowing-down, obtaining thus (new) points a, aD , b, bD and (new) curves A, AD , B, B D , c, cD , W, J on Y . The self-intersection numbers of these 8 curves are now, respectively, in this order: 4, 4, 0, 0, 0, 0, 0. Because the curves B, B D , c, cD are pairwise disjoint and intersect transversally each in exactly one point each of the two curves W and J, which are disjoint, we see that Y is isomorphic to P1 × P1 , with coordinates (s, t), the curves B, B D , c, cD (resp. W, J) being ﬁbres of the ﬁrst (resp. second) projection. Notice that W (resp. J) meets B and B D transversally at b = hD and bD = h (resp. at a = v and aD = v D ), and that A (resp. AD ) is tangent to B and B D at a = v and bD = h (resp. is tangent to B and B D at b = hD and aD = v D ). This is consistent with the fact that A (resp. AD ) meets transversally c and cD at two points exactly D (d± and d± i ), so that A and A are of bidegree (1, 2), since they both meet W and J at one point only, transversally. To prove Proposition 5.1, it is now suﬃcient to ﬁnd the coordinates (s, t) of the D D D 8 points d± , d± i , a, a , b, b in suitable coordinates, since S is indeed obtained from D D Y by blowing-up the points a, a , b, b twice, as said, and the points d± , d± i once. The coordinates we choose on J are the ones obtained from the Weierstrass P -function (multiplied by a suitable constant such that P ( 12 ) = 1) on the elliptic curve E×{ 21 } on E×E which covers twice the curve J under the rational projection map from E × E to Y obtained taking ﬁrst the strict transform in AD , and then by composing the map AD → S D with the above blow-downs. This (suitably modiﬁed) Weierstrass P -function thus maps the points 1 1 1 i 1 1+i 1 (0, ), ( , ), ( , ), ( , ) 2 2 2 2 2 2 2 on E × E to the points (∞, 0), (∞, 1), (∞, −1), and (∞, 0) on Y ∼ = P1 × P1 D D respectively. The curves B, B , c and c have now equations s = ∞, s = 0, s = 1 and s = −1 respectively. The curves W and J have equations t = 0 and t = ∞ respectively. It thus only remains to determine the coordinates of the points 4 points d± and d± i , once coordinates on B have been chosen (presently, only the points with coordinates 0, ∞ are given, the coordinates thus still depend on a multiplicative constant). These 4 points are the intersections points of the curve A with the curves c and cD respectively. Notice that A is the graph of a map of degree two from B → J, since of bidegree (1, 2). The second projection of Y ∼ = J × B thus identiﬁes B and A. It is thus suﬃcient to choose coordinates on A which send the point a = v to 0 and b = h to ∞. Now A is naturally identiﬁed with the line of tangent directions to the point (0, 0) in E × E. Under this identiﬁcation, we have 6 natural tangent directions corresponding to the tangent directions of the

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elliptic curves E ×{0}, D± , Di± , and {0}×E, with natural coordinates 0, ±1, ∓i, ∞ respectively (because the map z → z −1 on P1 ﬁxes ±1 and maps ±i to ∓i). This gives us the claimed coordinates for the remaining 4 points d± and d± i . Indeed, on S D , the curve D+ meets c and A transversally, this last curve at the point of coordinate t = 1 (which is the tangent direction of D+ at (0, 0) in E × E). Its image in Y is the point d+ , intersection of A and c, c being the curve of equation s = P ( 21 ) = 1. The argument for the points d− , d± i is similar.

References [GHS03]

T. Graber, J. Harris, and J. Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57-67 (electronic), 2003.

[Mi-Mo86] Y. Miyaoka, S. Mori. A numerical criterion for uniruledness. Ann. Math. 124 (1986), 65-69. [U75]

K. Ueno. Classiﬁcation Theory of Algebraic Varieties and Compact Complex Spaces. LNM 439.

Fr´ed´eric Campana E-mail: [email protected]

Special Orbifolds and Birational Classiﬁcation: a Survey Fr´ed´eric Campana Contents 1 Introduction: The Decomposition Problem 2 The 2.A 2.B 2.C 2.D 2.E 2.F

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Three ‘Pure’ Geometries The trichotomy: numerical version . . . . . . The trichotomy: birational version . . . . . . A remark on the LMMP . . . . . . . . . . . . Rational connectedness and κ+ = −∞ . . . . Birational stability of cotangent bundles . . . The decomposition problem: a failed attempt

3 Geometric Orbifolds 3.A Geometric orbifolds . . . 3.B Orbifold invariants . . . 3.C Orbifold morphisms . . 3.D Orbifold birational maps

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4 Orbifold Base of a Fibration 4.A Orbifold base . . . . . . . . . . . . 4.B Birational (non-) invariance . . . . 4.C The diﬀerential sheaf of a ﬁbration 4.D Fibrations of base-general type . . 4.E Orbifold ﬁbres and suborbifolds . . 4.F Relative diﬀerentials . . . . . . . .

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6 Special Orbifolds 6.A Deﬁnition and main examples . . . . . . . . . . . . . . . . . . . . . 6.B Criteria for special orbifolds . . . . . . . . . . . . . . . . . . . . . . 6.C Special surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 The Decomposition of the Core 8.A Weak orbifold rational quotient . . . . . . . . . . . . . . . . . . . . 8.B The conditional decomposition c = (M ◦ r)n . . . . . . . . . . . . .

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9 Orbifold Rational Curves 9.A Minimal orbifold divisors . . . . . . . . . . . . . . . . 9.B Orbifold rational curves . . . . . . . . . . . . . . . . . 9.C Orbifold uniruledness and rational connectedness . . . 9.D Fano orbifolds . . . . . . . . . . . . . . . . . . . . . . . 9.E Orbifold uniruledness and canonical dimension . . . . 9.F Global quotients: lifting and images of rational curves 9.G Δ-rational curves vs ΔQ -rational curves . . . . . . . .

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10 Some Relations with LMMP and the Abundance Conjecture 11 Some Conjectures 11.A Lifting of properties . . . . . . . . . . . . 11.B The case of special orbifolds . . . . . . . . 11.C The general case . . . . . . . . . . . . . . 11.D Families of canonically polarised manifolds

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13 Classical versus ‘Non-Classical’ Multiplicities

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14 An Orbifold Version of Mordell’s Conjecture

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1. Introduction: The Decomposition Problem This text is an expansion of a talk given at the Schiermonnikoog Conference on Birational Classiﬁcation, organised in May 2009 by C. Faber, E. Looijenga and G. van der Geer. It is intended to give an exposition of the main content of [Ca 04] and [Ca 07] (with some additional topics in §4.F, §10 or developments, as in §11.D), as brieﬂy and simply as possible, but essentially skipping the proofs. Some topics studied in [Ca 04] and [Ca 07] have not been included here (such as orbifold versions of fundamental groups, universal covers and function ﬁelds). We shall show how to decompose, by functorial and canonical ﬁbrations, arbitrary n-dimensional complex projective1 varieties X into varieties (or rather ‘geometric orbifolds’) of one of the three ‘pure’ geometries determined by the ‘sign’ (negative, zero, or positive) of the canonical bundle. As these decompositions are birationally invariant, birational versions, rather, of these ‘pure’ geometries, based on the ‘canonical’ (or ‘Kodaira’) dimension will be considered. This ‘pure’ trichotomy reﬁnes a more fundamental new dichotomy: ‘special’ opposed to ‘general type’ (the usual notion). ‘Special’ turns out to be a suitable ‘orbifold’ combination of the ﬁrst two pure geometries (canonical bundle negative or zero). More precisely, a variety X is ﬁrst decomposed2 , via a birationally unique 1 Although the geometric results apply to compact K¨ ahler manifolds without change, we consider here for simplicity this special case only. 2 Everything works for orbifolds (X|Δ) as well.

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ﬁbration c : X → C(X) (its ‘core’) into its antithetical parts: ‘special’ (the ﬁbres) and ‘general type’ (the ‘orbifold base’)3 . This is the new and most fundamental result of [Ca 04] and of the present text. Next, the ‘core’ can be decomposed4 as a composition c = (M ◦ r)n of orbifold versions of the weak ‘rational quotient’5 r, and the Moishezon-Iitaka ﬁbration M . When X is ‘special’, C(X) is a point, and X is thus a tower of ﬁbrations with ﬁbres of the ﬁrst two geometries. A crucial feature of these decompositions is indeed that, in order to deal with multiple ﬁbres of ﬁbrations, they need to take place in the larger category of ‘geometric orbifolds’ (X|Δ). These are ‘virtual ramiﬁed covers’ of varieties, which ‘virtually eliminate’ multiple ﬁbres of ﬁbrations. Although formally the same as the ‘pairs’ of the LMMP (see [KaMaMa 87], [KM 98], [BCHM 06] and the references there), they are here fully geometric objects equipped with the usual geometric invariants of varieties, such as sheaves of (symmetric) diﬀerential forms, fundamental group, Kobayashi pseudometric, integral points, morphisms and rational maps. One expects theorems on birational properties of projective varieties to extend, together with their proofs (modulo some ‘orbifold’ adaptations) to the larger ‘orbifold’ category. This is illustrated by several examples below, the most important one being the weak positivity of direct images of pluricanonical sheaves in the ‘orbifold’ context. These ‘orbifold’ extensions are essential in the applications. In the ‘orbifold’ category (but not in the category of varieties), most of the basic properties of the ﬁrst two geometries are expected to be preserved by ‘extensions’ (i.e. by the total space of a ﬁbration, if satisﬁed by both the base and the general ﬁbre). This naturally leads to conjecturing, among many other things, that ‘special’ orbifolds have an almost abelian fundamental group, and are exactly the ones with vanishing Kobayashi pseudometric or potentially dense set of rational points if deﬁned over a number ﬁeld. Also, conjecturally, the ‘core’ should split the arithmetic and C-hyperbolic properties of ‘orbifolds’ in their two antithetical parts (‘special’ vs ‘general type’). An implicit underlying theme of the considerations below is the ‘birational stability’ of the cotangent bundle of a projective manifold. The question, vaguely formulated, is: to which extent is the amount of holomorphic sections of the symmetric powers on ΩpX controlled by those of the pluricanonical bundles? Precise formulations are given in §2.E. The expected answer is that these cotangent bundles are birationally stable, unless the manifold is uniruled, in which case the birational stability is restored on the ‘rational quotient’ of X. A similar answer is expected for orbifolds (see Conjecture 5.12). We show in §10 how to deduce these conjectures from standard conjectures of the LMMP. In this survey, many technicalities concerning the orbifold category have been skipped. The details and proofs are in [Ca 04] and [Ca 07], where other related themes are treated. The results presented below on orbifold birational equivalence need to be completed on several essential points, and extended. See Remarks 4.6 3 The term ‘orbifold’ is used by Deligne-Mostow in almost the same sense as here. See footnote in 3.1 below. The term ‘orbifold pair’ might possibly make the link to LMMP clearer. 4 Using an ‘orbifold’ version of Iitaka’s C n,m -conjecture. 5 Also called ‘MRC ﬁbration’.

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and 3.8. Several possibly accessible foundational questions remain unanswered. Essentially all of the deﬁnitions and results here apply to ‘smooth orbifolds’ only. It is essential to extend these to the larger category of ‘log-canonical’ pairs of the LMMP. The investigations described below have been deeply inﬂuenced by K. Ueno’s book [U75].

2. The Three ‘Pure’ Geometries We will now introduce the three ‘pure’ geometries which deﬁne the ‘elementary’ objects of our classiﬁcation theory. These ‘pure’ geometries are deﬁned in two versions: ﬁrst, ‘numerical’, according to the ‘sign’ of the canonical bundle, then, ‘birational’, according to the reﬁned ‘canonical’ dimension. The deﬁnitions can be extended to ‘smooth geometric orbifolds’ in the obvious way, but most known properties have not been shown to extend to this broader situation. 2.A. The trichotomy: numerical version. The canonical divisor KX of a manifold X has emerged as the major invariant of classiﬁcation6 . Even to such an extent that one could almost deﬁne algebraic geometry as aiming at “converting positivity (or negativity) of KX into geometry, topology and arithmetic of X”. However, when the canonical bundle has generically negative directions, the canonical bundle is not suﬃcient, by itself, to control all directions of the cotangent bundle. See §2.E below. The three elementary classes are thus the cases where KX is either anti-ample, or (numerically) trivial, or ample, which we will respectively denote by: KX < 0, KX ≡ 0 and KX > 0. For these three classes of manifolds some of the qualitative properties of curves are known to hold in higher dimension, the others being conjectural, which is indicated by the sign “?”. The symbol “??” means that no general conjecture is presently even formulated, while ‘essentially’ means: ‘outside of a proper algebraic subset’. We denote by dX the Kobayashi pseudometric of X (deﬁned in 3.5 below), and by X ∗ (k) the Zariski closure of the set of k-rational points of X. KX <0 ≡0 >0

π1 (X) {1} almost abelian ??

dX ≡0 ≡ 0? ‘essentially’ a metric ?

X ∗ (k) X? X? ‘essentially’ ﬁnite ?

We see that essentially nothing is known in higher dimensions on the arithmetic side, and also for varieties of general type. The vanishing of the Kobayashi pseudometric on a Fano manifold follows from its rational connectedness (see below), which is one of the main applications of Mori theory ([Ca 92], [KMM 92]). 6 Only recently it seems. First in the consideration of Enriques plurigenera and in the classiﬁcation of surfaces by Kodaira and Shafarevich et al., and then in the three-dimensional classiﬁcation initiated by Mori, Kawamata and Shokurov.

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2.B. The trichotomy: birational version. We deﬁne now the three pure geometries as follows (except for 1, this is Ueno’s trichotomy in [U 75], §11): 1. κ+ (X) = −∞ (see Deﬁnition 2.3 below), which is conjecturally equivalent (see Conjecture 2.2) to rational connectedness. 2. κ(X) = 0, 3. κ(X) = dim X. In analogy to the case of curves and the numerical version of the trichotomy, the expected properties (all of them conjectural) are as follows: κ κ+ (X) = −∞ κ(X) = 0 κ(X) = dim X

π1 (X) {1}? almost abelian?

dX ≡ 0? ≡ 0? essentially > 0 ?

X ∗ (k) X? X? essentially ﬁnite ?

2.1. Remark. Let us give some reasons supporting these conjectures. 1. If Conjecture 2.4 holds, a manifold with κ+ (X) = −∞ is rationally connected, so its Kobayashi pseudometric vanishes, and X is simply connected. 2. If κ(X) = 0, the (conjectural) minimal model program implies that X is birational to a mildly singular7 variety X D such that the canonical divisor KX " is numerically trivial. Singular versions of the Bogomolov decomposition theorem should imply that the fundamental group is almost abelian. The “classiﬁcation” of varieties with numerically trivial canonical divisor suggests that the Kobayashi pseudometric vanishes. For curves and surfaces this is known. In the case of “Hyperk¨ahler” manifolds, only weaker versions are known, using twistor spaces. But the proof suggests this vanishing. The case of Calabi-Yau manifolds (simply connected, with h(0,2) = 0) is much more conjectural. The computation of GromovWitten invariants on some examples shows however the existence of inﬁnitely many rational curves. 2.C. A remark on the LMMP. The aim of the MMP is actually to reduce the ‘birational’ version above to the ‘numerical’ version by constructing ‘minimal models’, and converting hypotheses of pseudo-eﬀectivity for adjoint line bundles by nefness ﬁrst, and then by semi-ampleness. This task naturally leads (for reasons apparently diﬀerent from the ones here)8 to introducing the very same ‘pairs’ (X|Δ) as below, which permits an inductive treatment on the dimension, by producing ‘log-canonical centers’, and to extend the MMP to the LMMP. However, while the LMMP considers only the canonical bundle KX + Δ of pairs, our approach leads us to equip them naturally with many other geometric invariants. 2.D. Rational connectedness and κ+ = −∞. The following is one of the central problems in birational classiﬁcation, and the ﬁrst step of the so-called ‘Abundance conjecture’: 7 Technically:

‘terminal’. the proofs of the Kawamata-Viehweg Theorem, as well as their approach to weak positivity of direct images of pluricanonical sheaves rely on cyclic covers, the same is implicitly true for Kawamata’s subadjunction theorem. 8 Although

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2.2. Conjecture. Let X be a projective manifold such that κ(X) = −∞. Then X is uniruled. This conjecture is known up to dimension 3 by the already classical work of S. Mori, Y. Kawamata, V. Shokurov and Y. Miyaoka. The class of projective manifolds such that κ(X) = −∞ contains all products P1 × Y , and so does not deﬁne a ‘pure’ geometry. Hence the following deﬁnition: 2.3. Deﬁnition. We deﬁne: κ+ (X) = −∞ if and only if κ(Y ) = −∞ for all positive-dimensional manifolds Y dominated by X. Here “Y is dominated by X” means that there exists a dominant rational map g : X ''( Y . More generally, for any X, we deﬁne κ+ (X) := max{κ(Y )}, for all positivedimensional Y dominated by X. Obviously, n ≥ κ+ (X) ≥ κ(X), for any X, and κ+ (X) = −∞ if X is rationally connected. Conversely: 2.4. Conjecture. A manifold X is rationally connected if κ+ (X) = −∞. Recall the existence of the following map r, the ‘rational quotient’ (or ‘MRC ﬁbration’ in [KMM 92]), which describes the κ+ = −∞ ‘part’ of any X. 2.5. Theorem. [Ca 81, KMM 92, GHS 03] Let X be a projective manifold, then there exists a unique rational map: rX : X ''( R(X) such that i) its ﬁbres are rationally connected; ii) R(X) is not uniruled. Remark. It easily follows from this theorem that Conjecture 2.2 implies Conjecture 2.4. Indeed, if rX : X ''( R(X) is not the constant map, R(X) is not uniruled, thus κ(R(X)) ≥ 0 by Conjecture 2.2, contradicting the assumption κ+ = −∞. 2.E. Birational stability of cotangent bundles. Let T (X) be the (birational) complex algebra of all ‘covariant’ holomorphic tensors (i.e. sections of ⊗Ω1X ) on the complex projective manifold X. This algebra contains fundamental information on the rational ﬁbrations f : X ''( Y (we give two examples below). The study of this algebra is diﬃcult, since it is deduced from the rank-n cotangent bundle of X. It is thus of central importance to deduce its qualitative structure from positivity properties of the rank-one canonical bundle alone, or in other words, to establish the ‘birational stability’ of Ω1X (in a precise sense given below). This is true in the numerical version, i.e. when the numerical properties of KX are considered (see below). This fails however completely when the birational invariants are considered, speciﬁcally when κ(X) = −∞. This property is much too weak to control all directions of the (co)tangent bundle, as shown by X = P1 × Y, where Y is any projective manifold. This stability should hold, however, under the reﬁned condition κ+ (X) = −∞, which considers the canonical bundles of all ‘quotients’ of X, such as Y when X = P1 × Y . When X is of ‘pure geometry’ in the ‘numerical version’, the canonical bundle determines the structure of T (X). Indeed, when X is Fano, and so rationally connected, this algebra reduces to C, the constants. When KX ≡ 0, Miyaoka generic semi-positivity9 implies that non-zero elements of T (X) do not vanish 9 Or,

alternatively, using Ricci-ﬂat metrics, the parallelism of these tensors.

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anywhere, which permits the explicit determination of T (X) by representation theory, and implies that this algebra is ﬁnitely generated (see [Ca 95] or [Pe 94] for the details of the argument). When KX is ample, the situation is not uniform, the symmetric part of the algebra above being possibly reduced to the constants (for hypersurfaces of the projective space, when n ≥ 2). But sections of higher jet bundles are expected to always exist in abundance; this is an essential issue in hyperbolicity questions. See, for example, [D 97],[DMR 08], where many more references are also given. The proofs of these results, which permit to control the rank-n bundle Ω1X by its determinant line bundle, use deep and indirect results: either Mori’s reduction to char p > 0, or Yau’s solution of the Calabi Conjecture. For general X of intermediate positivity, these algebras (which are birational invariants, possibly even deformation invariants) have apparently not been investigated. For the ‘pure geometries’ in their ‘birational versions’, essentially nothing is proved, although the same behaviour as in the ‘numerical’ version is expected. Indeed, if Conjecture 2.4 holds, then T (X) = C when κ+ (X) = −∞ as well. When κ(X) = 0, the Abundance conjecture implies as above that T (X) should have the same structure as when KX ≡ 0 (see [Ca 95]). Bounding the positivity of Ω1X also permits to control the ﬁbrations on X and other invariants, such as even π1 (X). More precisely, we formulate in Conjectures 2.7 and 2.10 below the expected ‘birational stability’ of Ω1X : 2.6. Deﬁnition. Let X be a projective complex connected manifold. Then one sets κ++ (X) := max {κ(X, L) : L ⊂ ΩpX a rank 1 coherent subsheaf }. (L,p>0)

Obviously one has: n ≥ κ++ (X) ≥ κ+ (X) ≥ κ(X). 2.7. Conjecture. If κ(X) ≥ 0, then κ++ (X) = κ(X). Moreover, if κ+ (X) = −∞ then κ++ (X) = −∞. More precisely: κ++ (X) = κ(R(X)) for any X, with R(X) being its ‘rational quotient’. In particular: κ++ (X) = κ+ (X) for any X. 2.8. Remark. Easy arguments (using the Moishezon-Iitaka ﬁbration) show that if the conjecture holds for κ = 0, it also holds when κ ≥ 0. The same arguments also apply in the orbifold case (see Conjecture 5.12). If f : X ''( Y is a non-constant ﬁbration, κ(Y ) = κ(X, f ∗ (KY )) ≤ κ++ (X). The conjecture above thus implies: κ(Y ) ≤ 0 if κ(X) = 0 (which follows from Iitaka’s Cn,m too). Bogomolov’s theorem ([Bog 79]) asserts that κ(X, L) ≤ p if L ⊂ ΩpX is a rank 1 coherent subsheaf, and that L = f ∗ (KY ) generically on X, for some unique f : X ''( Y , if equality holds. However, Y needs then not be of general type. The diﬀerence between L and f ∗ (KY ) lies in the multiple ﬁbres of f , and the ‘orbifold base’ (Y |Δf ) is of general type in this case, as will be seen below (in the more general ‘orbifold’ context). The very same results and expectations hold also for ‘smooth geometric orbifolds’ (see Conjecture 5.12 below).

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Another similar birational invariant which has implications on the structure of the universal cover is: 2.9. Deﬁnition. [Ca 95] Let X be a projective complex connected manifold. Then: κ+ (X) := max{F ,p>0} {κ(X, det(F))}, where F ⊂ ΩpX is a coherent subsheaf. Similarly to 2.7: 2.10. Conjecture. [Ca 95] If κ(X) ≥ 0 then κ+ (X) = κ(X). If κ+ (X) = −∞ then X is rationally connected. More precisely: κ+ (X) = κ(R(X)) for any X. This conjecture is established when κ+ (X) = n (see [CPe 05]). When κ(X) = 0, it implies Ueno’s Conjecture that h0 (X, ΩpX ) ≤ (np ) if κ(X) = 0, which is still open. Using L2 -theory, Atiyah’s index theorem, and Gromov’s Poincar´e series [Gr 91], it is shown in [Ca 95] that: 2.11. Theorem. Assume that χ(OX ) K= 0. Then: γ(X) ≤ κ+ (X). Here γ(X) := dim(Γ(X)), where γX : X ''( Γ(X) is the ‘Gamma-reduction’ (or ‘Shafarevich map’) of X, with general ﬁbres the largest subvarieties in X whose fundamental group has ﬁnite image in π1 (X). The ﬁbre-dimension of γX is also the dimension of the largest compact connected analytic subset through a general point of the universal cover of X. This result implies in particular that π1 (X) = {1} if κ+ (X) = −∞, that π1 (X) is ﬁnite if κ+ (X) = 0 and χ(OX ) K= 0, and that X is of general type if γ(X) = n and χ(OX ) K= 0 ([CPe 05]). Observe that Abelian varieties have κ = κ+ = n, and γ(X) = n, so that the condition χ(OX ) = K 0 is essential in the last two statements. An example of a threefold X of general type with γ(X) = n = 3, and χ(OX ) = 0 is given in [EL 95]. 2.F. The decomposition problem: a failed attempt. Any curve belongs to one of the three pure geometries. In dimension at least two, this is no longer true, and ﬁbrations (i.e. dominant rational maps f : X ''( Y with connected general ﬁbre) are needed to decompose arbitrary manifolds X into pieces of ‘pure geometry’. The two classical maps: ﬁrst r : X → R(X) (the “rational quotient” or “MRC”ﬁbration, see [Ca 92] or [KMM 92]), and M = MX : X → M (X) (the MoishezonIitaka ﬁbration, if κ(X) ≥ 0) seem, at ﬁrst sight, to provide a solution to this decomposition problem. Indeed, the ﬁrst one eliminates the RC ‘part’ of X, and the second its κ = 0 ‘part’. Since r has a non-uniruled base R(X), it has conjecturally κ(R(X)) ≥ 0 (by Conjecture 2.2), so that M ◦ r : X → M (R(X)) should be well-deﬁned for any X. However, M (R(X)) is not of general type in general, and one needs to iterate and consider (M ◦ r)n : X → M Rn (X) to reach a base M Rn (X) of general type (possibly a point). The problem is that ‘parts’ of general type can be hidden in the seemingly ‘general type-free’ ﬁbres of (M ◦ r)n . The simplest example is given in

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2.12 below. This (M ◦r)-process thus reduces the dimension, but does not describe the structure of general X. Notice that the LMMP, which aims at producing a ‘numerical’ version of the maps M and r (converting, for appropriate adjoint Q-line bundles, the word ‘pseudo-eﬀective’ into ‘nef’ ﬁrst, and next into ‘semi-ample’) does not consider this question. As we shall see, this decomposition process works, but only in the ‘orbifold category’ deﬁned below. Indeed, as shown by Example 2.12 below, the main problem is that the couple (Xy , Y ), where Xy is the general ﬁbre and Y the base of some ﬁbration f : X → Y , does not determine the qualitative geometry of X, even when f is one of the natural ﬁbrations of the minimal model (or of the Moishezon-Iitaka) program. This failure is due to the presence of multiple ﬁbres, and disappears in the category of ‘geometric orbifolds’, which consists precisely in encoding this data. Surprisingly, this single addition is suﬃcient to correct the (M ◦ r)n decomposition above. Working in the category of geometric orbifolds is thus not only necessary, but also suﬃcient to solve the decomposition problem. Let us illustrate this failure with the following very simple example. 2.12. Example. Let C be a hyperelliptic curve, and let E be an elliptic curve (both deﬁned over some number ﬁeld k, say). Let i : C → C be the hyperelliptic involution that exchanges the ﬁbres of the morphism C → P1 and let t : E → E be a translation of E of order 2, thus t has no ﬁxed point. The quotient map u : X D := E × C → X := (C × E)/mi × tC is thus ´etale, and κ(X) = 1. We thus get a commutative diagram, in which the vertical arrows are also the Moishezon-Iitaka ﬁbrations: XD = C × E

u (2:1) ´ etale

MX "

# C

v (2:1) ramiﬁed

! X = X D /mi × tC #

MX

! P1 = C/miC

Because π1 (X), dX and X(k) are essentially invariant under ﬁnite unramiﬁed covers, these invariants coincide essentially with those of X D , and thus diﬀer radically from those expected by considering the couple (Xy , Y ) = (E, P1 ) of generic ﬁbre and base of MX : X → P1 . Seeing this ﬁbration as a ‘twisted product’ E ×P1 , one would expect X to have an almost abelian fundamental group, dX ≡ 0, and X(k) Zariski dense (after a ﬁnite extension). The qualitative geometry of X can be recovered by looking at X only, since X D is nothing but the normalisation of the ﬁbre product X ×P1 C and C is simply a ramiﬁed cover of P1 which ramiﬁes at order two exactly over the points p ∈ P1 over which MX has a double ﬁbre. Under the normalised ﬁbre product above, these multiple ﬁbres are thus eliminated. We will now generalise this construction. Let f : X → Y be a ﬁbration. Let Δf be the Q-divisor on Y uniquely deﬁned by the multiple ﬁbres of f , in such a way that base-changing f by a local ﬁnite cover Y D → Y ramifying ‘exactly’ over

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Δf eliminates in codimension one the multiple ﬁbres of the ﬁbration f : X → Y . Since a ﬁnite ramiﬁed cover Y D → Y such as C → P1 above does not exist in general globally, we need to work directly with the pair (Y |Δf ).

3. Geometric Orbifolds We give only the deﬁnitions needed for the considerations below. The other deﬁnitions are in [Ca 07]. Throughout this section we will denote by X a smooth projective manifold of dimension n. 3.A. Geometric orbifolds. 3.1. Deﬁnition. Let Y be a normal connected % complex projective variety. An orbifold divisor is an eﬀective Q-divisor Δ = D⊂Y (1 − 1/m(D))D, where: i) the sum ranges over all prime divisors on Y , ii) m(D) ∈ (Q ∩ [1, +∞[) ∪ {+∞}, iii) m(D) = 1 for all but a ﬁnite number of D’s. A ‘geometric orbifold’ (or simply: ‘orbifold’)10 is a couple (Y |Δ) where Y is smooth11 projective and Δ an orbifold divisor on Y ; it is said to be ﬁnite (resp. integral, resp. logarithmic) if for all D ⊂ Y , one has: m(D) < +∞ (resp. m(D) ∈ (Z ∪ +∞), resp. m(D) = +∞). The ‘support’ Supp(Δ) = iΔA of Δ is the (ﬁnite) union of all D’s such that m(D) > 1. The geometric orbifold (Y |Δ) is said to be ‘smooth’ if Y is smooth, and if Supp(Δ) is a divisor of normal crossings. When Δ = 0 (resp. when Δ is logarithmic) the orbifold (X|Δ) is identiﬁed with X (resp. with the quasi-projective variety U := X − Δ). We deﬁne a lattice order on the set of orbifold divisors on X by writing ΔD ≥ Δ if (ΔD − Δ) is eﬀective. Geometric orbifolds (Y |Δ) with ﬁnite multiplicities interpolate between proper or compact orbifolds (when Δ = 0), and open orbifolds (when the multiplicities are all inﬁnite). When moreover integral, they may be considered as virtual covers of Y ramiﬁed over each divisor D of Y with multiplicity mΔ (D) := m(D) in the notation above. Geometric orbifolds thus coincide with the ‘log-pairs’ (with rational coeﬃcients) of the LMMP. Notice that a smooth orbifold is log-canonical, and klt if ﬁnite. The origin and main source of geometric orbifolds here are the ‘orbifold bases’ of ﬁbrations (see §4 below). 10 In [DM 93, §14, pp. 135-141], this term is employed in a sense which is essentially equivalent to ours in the ﬁnite, integral, smooth case. This reference was pointed out to me by F. Catanese. 11 Being normal and Q-factorial is actually suﬃcient for most of the deﬁnitions given here.

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3.B. Orbifold invariants. The most fundamental one is the following: 3.2. Deﬁnition. Assume Y to be smooth (or KY to be Q-Cartier). The canonical bundle of the orbifold (Y |Δ) is deﬁned as KY |Δ = KY + Δ, and the canonical dimension of (Y, Δ) is κ(Y |Δ) := κ(Y, KY |Δ ). One thus has: dim Y ≥ κ(Y |Δ) ≥ κ(Y ). Other important invariants are, when (X|Δ) is smooth: • The locally free sheaves S N Ωq (Y |Δ) for any N, q ≥ 0, when (Y |Δ) is smooth, and p-dimensional. They agree with SymN (ΩqY ) if Δ = 0, and with SymN (ΩqY (log(D))) if Δ = Supp(Δ) = D is logarithmic. In general they interpolate between these two cases. In local analytic ‘adapted’ coordinates z = (z1 , . . . , zp ), in which the support of Δ consists of coordinate hyperplanes, the sheaf S N Ωq (Y |Δ) is generated as an dz(JE ) dz(J) OY -module by the elements z(J) := z ?k/m8 ⊗\=N \=1 z(J ) , where: E

1. the J\ are increasing subsets of {1, . . . , p} of cardinality q, lexicographically dz dz ordered in increasing order, and z(J(JE)) := ∧j∈JE zjj . E

2. kj is the number of ]’s such that j ∈ J\ , for each j ∈ {1, . . . , p}. ?kj /mj 8 3. z ?k/m8 := Πj=p , where mj is the Δ-multiplicity of the hyperplane j=1 zj with equation zj = 0. dz(J) −3kj (1−1/mj )@ \=N Note that z(J) := Πj=p ) ⊗\=1 dz(JE ) , for any (J). j=1 (zj The following lemma will be used in the proof of Lemma 10.6.

3.3. Lemma. With the notations above, assume that, for any j ∈ {1, . . . , p}, one has: mj > 1. Then, for any index (J) = (J1 , . . . , Jq ) above, there exists some j = p j((J)) such that \kj (1 − 1/mj )L > 0 if N ≥ q(1−1/m) , where m := inf j {mj } > 1. % Proof: Assume, by contradiction,%that kj (1 − 1/m) < 1 for each j. Since j kj = N q, we get: N q (1 − 1/m) ≤ j kj (1 − 1/mj ) < p, and the claim. • The sheaves of holomorphic tensors of type Tsr on (Y |Δ) are deﬁned similarly. See [Ca 07] for more details, and [JK 09] for some additional properties. In the proof of Lemma 10.7, we shall need a relative version, of independent interest, of the sheaves of diﬀerentials deﬁned above. See §4.F below. When Δ is integral, one can also naturally deﬁne (see [Ca 07] for details): • The fundamental group: π1 (Y |Δ), and the universal cover when (Y |Δ) is moreover smooth. • The Kobayashi pseudometric dY |Δ (see Example 3.5 below). • The notion of integral points (Y |Δ)(Ok,S ) if k is a number ﬁeld over which (Y |Δ) is deﬁned, with ring of integers Ok , and a ﬁnite set of places S. (This depends on the choice of a model of (Y |Δ) over Ok ). See [Ab 07] for a detailed exposition. 3.C. Orbifold morphisms. Let f : X → Y be a regular map between projective normal varieties. Assume Y to be smooth (or Q-Cartier). Let ΔX and ΔY be orbifold divisors on X and Y respectively.

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% For any prime Weil divisor E on Y , let: f ∗ (E) := D⊂X tE,D D, the sum running over all prime divisors on X. Thus tE,D > 0 if and only if f (D) ⊂ E. We write mX (D) := mΔX (D), and similarly for E, Y . 3.4. Deﬁnition. We say that f : (X|ΔX ) → (Y |ΔY ) is an orbifold morphism if i) f (X) is not contained in Supp(ΔY ). ii) For any E, D as above, if tE,D > 0, then tE,D mX (D) ≥ mY (E). We shall say that the orbifold morphism f : (X|ΔX ) → (Y |ΔY ) is a ‘classical’ orbifold morphism if (Y |ΔY ) and (X|ΔX ) are integral, and if moreover, in the second condition above “tE,D mX (D) ≥ mY (E)” is replaced by “mY (E) divides tE,D mX (D)” 3.5. Example. Assume (Y |ΔY ) is smooth. A holomorphic map h : D → Y such that f (D) is not contained in Supp(ΔY ) deﬁnes an orbifold morphism (resp. a classical orbifold morphism) h : D → (Y |ΔY ) if, for any z ∈ D such that f (z) ∈ Supp(ΔY ), the order of contact of f (D) with every branch E of Supp(ΔY ) at f (z) is at least (resp. is a multiple of) mY (E). (The notion of orbifold morphism does not need the projectivity of X or Y ). This leads to a deﬁnition of the Kobayashi pseudo-metric dY |ΔY for a smooth orbifold: this is the largest pseudo-metric δ on Y such that h∗ (δ) ≤ dD for any orbifold morphism h : D → (Y |ΔY ), with dD the Poincar´e metric on D. If (Y |ΔY ) is integral, we can deﬁne similarly the ‘classical’ Kobayashi pseudo-metric d∗(Y |ΔY ) on (Y |ΔY ), by replacing the orbifold morphisms h : D → (Y |ΔY ) above by their ‘classical’ analogues. We have natural inequalities: d∗(Y |ΔY ) ≥ d(Y |ΔY ) ≥ dY . When Y is a curve, we have: d∗(Y |ΔY ) = d(Y |ΔY ) ([Rou 06], answering a question in [CW 05]). The notions of morphisms and diﬀerential forms are compatible12 : 3.6. Theorem. ([Ca 07], 2.7 and 2.12.) Let f : X → Y be holomorphic, with X, Y smooth, and ΔX , ΔY as above. Assume that f (X) is not contained in Supp(ΔY ). The following conditions are then equivalent: i) f : (X|ΔX ) → (Y |ΔY ) is an orbifold morphism. ii) f ∗ (S N Ωp(Y |ΔY ) ) ⊂ S N Ωp(X|ΔX ) , ∀p, N ≥ 0. iii) If (X|ΔX ) is integral: for any orbifold morphism h : D → (X|ΔX ), f ◦ h : D → (Y |ΔY ) is also an orbifold morphism. 12 When one considers orbifolds, even log-terminal, with X being Q-factorial, but not factorial, the deﬁnition of an orbifold morphism might need to take into account the non-factoriality of X, by introducing a suitable factor of local non-factoriality. The simplest example, pointed out by the referee, being the cone over the conic equipped with two lines of the ruling with multiplicity 2 as Δ, and blown up at the vertex, exhibits a diﬀerence between the usual orbifold theory, where the exceptional divisor has multiplicity 1, and the deﬁnition above, which gives multiplicity 2.

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3.D. Orbifold birational maps. The following deﬁnitions need to be generalised to the larger class of log-canonical and klt orbifolds (instead of smooth ones). 3.7. Deﬁnition. Let f : X → Y be a birational map between smooth projective manifolds equipped with orbifold divisors ΔX , ΔY . We say that f : (X|ΔX ) → (Y |ΔY ) is an elementary birational orbifold morphism (an ‘ebom’) if: 1) (Y |ΔY ) and (X|ΔX ) are smooth, 2) f : (X|ΔX ) → (Y |ΔY ) is an orbifold morphism, 3) f∗ (ΔX ) = ΔY . The smooth orbifolds (X|ΔX ) and (Y |ΔY ) are said to be birationally equivalent if they can be connected by a chain of ‘eboms’. From Theorem 3.6, we deduce that the spaces of global sections of the sheaves S N (Ωp∗ ) are orbifold birational invariants, since these sheaves are locally free. 3.8. Remark. In general, two birational smooth orbifolds are, however, not birationally dominated by a third one (see [Ca 07], 2.32 and 2.33). This is the source of considerable technical, even possibly conceptual, diﬃculties. 3.9. Remark. Two logarithmic orbifolds are birationally equivalent in the sense above if and only if their open parts (X − Δ) are (algebraically) isomorphic. Thus birational invariants in the orbifold category produce (many new) invariants of quasi-projective varieties.

4. Orbifold Base of a Fibration 4.A. Orbifold base. In the sequel, (f |Δ) : (X|Δ) → Y will denote a smooth orbifold (X|Δ), with X projective and n-dimensional, together with a ﬁbration f : X → Y , Y being p-dimensional. % 4.1. Deﬁnition. Let D ⊂ Y be a prime divisor. Then f ∗ D = k tk Ek + R, where f (Ek ) = D and codimY (f (R)) ≥ 2, i.e. R is an f -exceptional divisor. We deﬁne m(f |Δ; D) := inf k {tk mΔ (Ek )} to be the multiplicity of the (f |Δ)-ﬁbre over a general point of D. % Set Δ(f |Δ) := D⊂Y (1 − 1/m(f |Δ; D))D. The ‘orbifold base’ of (f |Δ) is deﬁned to be (Y |Δ(f |Δ) ). When Δ = 0, we simply write m(f ; D) and Δf . Remark. 0. The (simple) reason for this deﬁnition is given in Remark 4.2 below. 1. Since the general f -ﬁbre is smooth, so irreducible and reduced, it is clear that there are at most ﬁnitely many prime divisors D such that m(f |Δ; D) > 1. Thus Δf (resp. Δ(f |Δ) ) is always an integral orbifold divisor (resp. when so is Δ). 2. One has Δ(f |ΔD ) ≥ Δ(f |Δ) if ΔD ≥ Δ.

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3. In general, f : (X|Δ) → (Y |Δ(f |Δ) ) is an orbifold morphism only in codimension one, because the multiplicities on the f -exceptional divisors on X are not taken into account in the deﬁnition, and may be too small. This is remedied (to a certain extent, suﬃcient for the purposes) by the consideration of ‘neat models’ in the next section. 4. The ‘classical’ (or ‘divisible’) notion does not coincide with our multiplicity above. With this notation, it is deﬁned by m∗ (f |Δ; D) := gcdk {tk mΔ (Ek )}. Thus m∗ (f |Δ, D) divides m(f |Δ, D), and (f |Δ) will have ‘more’ multiple ﬁbres than in the classical sense, in general. 5. There are several main reasons for considering the inf, instead of the gcd multiplicities: compatibility with diﬀerentials (see 3.6), canonical dimension of orbifold bases (see 4.10). Also for hyperbolicity reasons, one can expect the Kobayashi pseudometric of (X|Δ) to be the lift from the one on the orbifold base of its ‘core’ (see §7) only with the inf multiplicities. 4.2. Remark. There is a composition rule13 for the base orbifold of the compof g sition g ◦ f two ﬁbrations X → Y → Z, namely: Δg◦f = Δ(g,Δf ) , when Δ = 0. This equality suggests the deﬁnition of the base orbifold of f : (X|ΔX ) → Y , given above. We refer to [Ca 04, ch.1.6] and [Ca 07, chap. 3] for more details. 4.B. Birational (non-) invariance. 4.3. Deﬁnition. Let f : X → Y and f D : X D → Y D be two ﬁbrations, and Δ, ΔD be orbifold divisors on X, X D respectively. Assume that u : (X D |ΔD ) → (X|Δ) is an ‘elementary birational orbifold morphism’ between smooth orbifolds. Then (f |Δ) is said to be ‘elementarily birationally equivalent’ to (f D |ΔD ), if there exists a birational map v : Y D → Y making commutative the diagram: XD

u

f"

# YD

v

!X # !Y

f

We say that (f |Δ) is ‘birationally equivalent’ to (f D |ΔD ), and write (f |Δ) ∼ (f D |ΔD ) if they are connected by a chain of elementarily equivalent ﬁbrations (fj |Δj ). 4.4. Lemma. If (f |Δ) is ‘elementarily birationally equivalent’ to (f D |ΔD ), we have the following properties: 1) v∗ (Δ(f D |ΔD )) = Δ(f |Δ), 2) κ(Y D |Δ(f D |ΔD )) ≤ κ(Y |Δ(f |Δ)), 3) κ(Y D |Δ(f D |ΔD )) = κ(Y |Δ(f |Δ)) if κ(Y |Δ(f |Δ)) ≥ 0. 13 On

suitable ‘neat’ birational models of f, g at least. See below for this notion.

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Remark. The inequality κ(Y D |Δ(f D )) ≤ κ(Y |Δ(f )) can be strict if κ(Y |Δ(f |Δ)) = −∞. In particular the canonical dimension of the orbifold base is not a birational invariant of ﬁbrations. The next deﬁnition remedies this situation, but using, in a ﬁrst step, a less computable invariant. This drawback will be cancelled using ‘neat ﬁbrations’ below. 4.5. Deﬁnition. Let f : X → Y be a ﬁbration, then we deﬁne the canonical dimension of (f |Δ) as κ(f |Δ) := inf (f " |Δ" )∼(f |Δ) {κ(Y D , KY " + Δ(f D |ΔD ))}. The canonical dimension κ(f |Δ) is now a birational invariant and we can extend the deﬁnition to any rational map f : X ''( Y with Y arbitrarily singular by resolving the singularities of Y and the indeterminacies of f . We shall now compute this canonical dimension diﬀerently in the next section 4.C. 4.6. Remark. The study of orbifold birational maps needs to be completed on several essential points. In particular: 1) They should be extended to the category of log-canonical and klt orbifolds. 2) It is not known whether the orbifold bases of ‘neat’ ﬁbrations are birational invariants (in the orbifold category), and this even for the three central ﬁbrations of the decompositions described below (the Moishezon-Iitaka ﬁbration, the ‘κ-rational quotient,’ and the ‘core’, see Remarks 8.2, 8.4 and 7.2). 4.C. The diﬀerential sheaf of a ﬁbration. 4.7. Deﬁnition. Let E be a locally free sheaf on a complex manifold X. Let F ⊂ E be a coherent subsheaf of E. The saturation F sat is the kernel of the map: E → (E/F)/(Tor E/F). It is also the largest subsheaf of E containing F, and identical with F at the generic point of X. 4.8. Deﬁnition. Let (X|Δ) be a smooth orbifold. Let f : X → Y be a holomorphic ﬁbration between projective manifolds, with p := dim Y . For all N ≥ 1, the canonical injection 0 → f ∗ ΩpY → ΩpX induces a injective morphism ∗ sat f ∗ (N KY ) = f ∗ ((ΩpY )⊗N ) → SymN (ΩpX ) and we deﬁne: LN (f |Δ) := (f (N KY )) ∗ N p to be the saturation of f (N KY ) in S Ω (X|Δ). N p By deﬁnition LN (f |Δ) is a coherent rank 1 subsheaf of S Ω (X|Δ)) and it is easily seen to be a birational invariant of the ﬁbration (f |Δ): elementary birational equivalences of ﬁbrations indeed induce isomorphisms at generic points, but not on global sections in general. Conditions for this are given in Theorem 4.10 below. Moreover, for N suﬃciently large and divisible, f ∗ (N (KY + Δ(f |Δ))) ⊂ LN (f |Δ) outside the f -exceptional divisors of X. This is an elementary local computation that we explain, to simplify notations, when Δ = 0, n = dim X = 2 and p = dim Y = 1. The map f is then given in suitable local coordinates (u, v) on X, and y on Y , by: f (u, v) = y = uk , near a generic smooth point of a component of multiplicity k of some ﬁbre X0 = f −1 (0) of f . Let m be the inf multiplicity of this ﬁber, so that k ≥ m. Thus dy f ∗ (N m (KY + Δ(f )) is generated, as an OX -module, by: f ∗ (( y(1−1/m) )⊗N m ) =

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k N m u(k−m) N (du)⊗N m . This gives the said inclusion, and equality when k = m, in this case. 4.9. Deﬁnition. A ﬁbration f : X → Y is ‘neat’ if there exists a birational morphism u0 : X → X0 such that X0 is smooth and if every f -exceptional prime divisor E ⊂ X is also u0 -exceptional. Neat ﬁbrations are constructed as follows: for any ﬁbration f : X → Y with X smooth, let Y D → Y be a birational morphism such that X ×Y Y D → Y D is ﬂat (which exists, by Raynaud’s ﬂattening theorem). A desingularisation X D → X ×Y Y D of the main component of X → X ×Y Y D , induces a neat ﬁbration f D : X D → Y D , by taking for u0 the natural projection u0 : X D → X. 4.10. Theorem. Let (X|Δ) be a smooth orbifold, and f : X → Y be a neat ∗ 0 ﬁbration, then: H 0 (X, LN (f |Δ) ) @ f (H (Y, N (KY + Δ(f |Δ) ))), for all large and suﬃciently divisible N . For any rational ﬁbration f : X ''( Y , we can now deﬁne: κ(f |Δ) := κ(Y D |Δ(f " |Δ" ) ), if f D : X D → Y D is a neat ﬁbration birationally equivalent to f , with u : (X D |ΔD ) → (X|Δ) an elementary birational orbifold morphism. The right hand side of the equality is independent of the choice of (X D |ΔD ) by Theorem 3.5 and the remark following Deﬁnition 3.6. Idea of proof: First assertion: The ﬁbration being neat, Hartog’s theorem shows that we do not change the space of sections on the left-hand side by allowing poles of arbitrary orders on sections of LN (f |Δ) along divisors of X which are f -exceptional, hence u0 -exceptional. We thus assume there are no such divisors. By the deﬁnition of LN (f |Δ) , we have then a natural inclusion: N (KY + Δ(f )) ⊂ N L(f |Δ) with cokernel supported on a divisor D of X ‘partially supported on ﬁbres of f ’, which means that if E ⊂ Y is an irreducible component of f (D), then f −1 (E) has an irreducible component DD not contained in D, and such that f (DD ) = E. An elementary general lemma now shows that, in this situation, the global sections of the two said sheaves coincide. Second assertion: when f is neat, it is an immediate consequence of the ﬁrst assertion. The general case follows from the fact that both sides are birational invariants of the ﬁbration. 4.11. Remark. By contrast, we do not know if the base orbifolds of two birationally equivalent neat ﬁbrations are birationally equivalent. This is an important question, and it seems quite diﬃcult. 4.D. Fibrations of base-general type. 4.12. Deﬁnition. A meromorphic ﬁbration (f |Δ) : (X|Δ) ''( Y is of basegeneral type if κ(f |Δ) = dim Y > 0, with (X|Δ) a smooth orbifold. (We shall abbreviate ‘base-general type’ to ‘general type’). Notation. We denote by F(X|Δ) the set of ﬁbrations of (base-)general type on (X|Δ) (up to birational equivalence).

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4.13. Deﬁnition. Let (X|Δ) be a smooth, projective orbifold. A ‘Bogomolov sheaf ’ on (X|Δ) is a rank one coherent, saturated, subsheaf L ⊂ ΩpX such that κ(X|Δ, L) = p > 0. 0

N,Δ

(X,(L By deﬁnition, κ(X|Δ, L) := limN >0 log(h log(N ) tion in S N Ωp (X|Δ) of the image of f ∗ (N.KY ).

))

, where LN,Δ is the satura-

Notation. We denote by Bog(X|Δ) the set of Bogomolov sheaves on X. Bogomolov sheaves are thus the rank one subsheaves of ΩpX of maximal Δ-positivity, by the next theorem, essentially due to Bogomolov. It rests on Deligne’s closedness of logarithmic forms on (X|iΔA). 4.14. Theorem. ([Bog 79], [Ca 07]) Let (X|Δ) be a smooth projective orbifold and L ⊂ ΩpX a rank 1 subsheaf. Then: κ(X|Δ, L) ≤ p, and if equality holds, there exists a meromorphic ﬁbration f : X ''( Y such that L = f ∗ ΩpY holds above the generic point of Y . Together with Theorem 4.10, Bogomolov’s theorem leads immediately to the following geometric description of Bogomolov sheaves: 4.15. Theorem. [Ca 07, Th´eor`eme 8.9] The map L : F(X|Δ) → Bog(X|Δ) given by [f ] → ? Lf is bijective. Remark. It is essential for the theorem to use ‘inf-multiplicities’, instead of ‘gcdmultiplicities’. Theorem 4.15 interprets geometrically the “Bogomolov sheaves”, which were only partially interpreted in 4.14. An important property of ﬁbrations of general type is: 4.16. Theorem. [Ca 07, Th´eor`emes 8.17-19] Let (f |Δ) : (X|Δ) ''( Y be a ﬁbration, with (X|Δ) smooth projective. Then f : X ''( Y is ‘almost holomorphic’ if either κ(f |Δ) = dimY is of general type, or if κ(f |Δ) ≥ 0 and (X|Δ) is ﬁnite. Recall that f is ‘almost holomorphic’ if its indeterminacy locus does not meet its general Chow-theoretic ﬁbre. 4.17. Example. When (X|Δ) is logarithmic, a rational map (f |Δ) : (X|Δ) ''( Y with κ(f |Δ) ≥ 0 need not be almost holomorphic. Take X = P2 , Δ the union of two lines meeting in a point a, and f : X → P1 the linear projection from a. 4.E. Orbifold ﬁbres and suborbifolds. Let (f |Δ) : (X|Δ) → Y be a ﬁbration from the smooth projective orbifold (X|Δ). For y ∈ Y , let Δy be the restriction of the divisor Δ to the ﬁbre Xy := f −1 (y) of f . From Sard’s theorem it follows that, for y ∈ Y generic, Δy is an orbifold divisor of Xy , and that the orbifold (Xy |Δy ) is smooth. We call it the generic orbifold ﬁbre of (f |Δ). Moreover, if (f D |ΔD ) : (X D |ΔD ) → Y D is a ﬁbration birationally equivalent to (f |Δ) : (X|Δ) → Y , it is easy to check that for some dense Zariski open subset U = U D in Y ∼ Y D , the orbifold ﬁbres of (f |Δ) and (f D |ΔD ) over y are birationally equivalent (in the orbifold sense).

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4.18. Remark. When (f |Δ) : (X|Δ) ''( Y is only rational, the generic orbifold ﬁbre of (f |Δ) is not well-deﬁned, up to birational orbifold equivalence (unless f is almost holomorphic). Even its canonical dimension is not well-deﬁned up to birational equivalence. Consider P2 , and the map g : X ''( P1 deﬁned by a pencil of conics through 4 points. Let u : X → P2 be the blow-up in these 4 points, with exceptional divisor E consisting thus of four (−1)-curves. Let Δ be the orbifold divisor on X obtained in attributing the multiplicity m to each component of E. The family of the strict transforms of the smooth members of the initial pencil of conics are ﬁbres of f := g ◦ u, and their orbifold canonical dimension is thus −∞ (resp. 0, 1) if m = 1 (resp. m = 2, m ≥ 3). 4.19. Deﬁnition. Let (X|Δ) be a smooth projective orbifold, and j : V → X the inclusion of an (irreducible) subvariety not contained in Supp(Δ). We deﬁne a ‘restriction’ of Δ to V as any smooth projective orbifold pair (W |ΔW ) together with a birational map g : W → V such that the composed map j ◦ g : (W |ΔW ) → (X|Δ) is an orbifold morphism. For any given birational map g : W → V , with exceptional divisor E, such that (E ∪ (j ◦ g)−1 (Supp(Δ))) is a divisor of normal crossings, there exists a smallest restriction (W |ΔW )min of Δ to V deﬁned on W . 4.20. Remark. 1. However, if h : W D → W is another birational morphism such that g ◦ h : D W → V satisﬁes the normal crossings condition above, it is not known presently if the minimal restriction (W D |ΔD )min of Δ to W D is an orbifold morphism14 . If this were so, the notion of minimal restriction would be birational (for given (X|Δ), V ). 2. However, the example in Remark 4.18 shows that, even for curves, and for generic ﬁbres of a rational ﬁbration, there is no possibility to deﬁne a birationally well-deﬁned notion of minimal restriction which is independent of the birational model of (X|Δ) on the corresponding strict transform of V . (Except for the generic member of a ‘base-point free’ covering family of subvarieties.) For this reason we give the following deﬁnition: 4.21. Deﬁnition. Let (X|Δ) be a smooth projective orbifold, and V ⊂ X a subvariety not contained in Supp(Δ). We say that the restriction of Δ to V has a (birationally invariant) property P if there is some (smooth) orbifold-birational model (X D |ΔD ) of (X|Δ) such that the (minimal) restriction of ΔD to the strict transform V D of V on X D has the property P . The properties P which will be important here will be: being ‘special’, or with κ = 0, or with κ+ = −∞. See below. 4.22. Proposition. Let Vt be the generic member of a covering family of subvarieties of X. Having property P is independent of the birational model of the (smooth and projective) (X|Δ). The ‘genericity’ here depends on the model. (This applies, in particular, to orbifold ﬁbres of rational ﬁbrations). 14 Except

when V is a curve, a case considered in more detail in §9.

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141

4.F. Relative diﬀerentials. Let f : X → Y be a ﬁbration (i.e. a regular surjective and connected map) between complex projective connected manifolds. Let Δ be an orbifold divisor on X, such that (X|Δ) is smooth. Let ΔY := Δf |Δ be the associated base orbifold divisor on Y . By suitable blow-ups of X and Y , and orbifold modiﬁcations of (X|Δ), we can assume that: 1) f is neat. 2) (Y |ΔY ) is smooth. 3) f : (X|Δ) → (Y |ΔY ) is an orbifold morphism. From the property (3) above, we deduce, for any pair of nonnegative integers, N, q, a natural map of sheaves f ∗ : S N (Ωq (Y |ΔY )) → S N (Ωq (X|Δ)). We shall derive suﬃcient conditions for the map above to be surjective at the level of global sections. The conditions bear on the sheaves S N (Ωq (Xy |Δy )), where Xy is a generic smooth ﬁbre of f , while Δy is the restriction of Δ to Xy . Notice that Sard’s (or Bertini’s) theorem implies that (Xy |Δy ) is a smooth orbifold. 4.23. Proposition. In the situation above the map f ∗ : H 0 (Y, S N (Ωq (Y |ΔY ))) → H 0 (X, S N (Ωq (X|Δ))) is surjective if, for any ﬁnite sequence (Nh , qh ), h = 1, . . . , t of pairs of positive integers, H 0 (Xy , S N1 Ωq1 (Xy |Δy )⊗· · ·⊗S Nt Ωqt (Xy |Δy )) = {0}. Proof. Step One. The ﬁrst (and longer) step consists in studying the natural ﬁltration of the sheaves of relative orbifold diﬀerentials. Let Xy be a generic ﬁbre as above, and let M be its (trivial) conormal bundle, of rank (n − p). There is a natural increasing ﬁltration F s , s = −1, 0, . . . , q of Ωq (X)|Xy with graded pieces equal to F s /F s−1 = Ωs (Xy ) ⊗ Λq−s (M ), for s = 0, 1, . . . , q. This ﬁltration induces an increasing ﬁltration Gt F s of SymN (F s ), for t = −1, 0, . . . , N , associated to the exact sequence 0 → F s−1 → F s → (F s /F s−1 ) → 0. It has thus graded pieces Gt F s /Gt−1 F s = Symt (F s−1 ) ⊗ SymN −t (F s /F s−1 ). Altogether we get a ﬁltration of SymN (Ωq (X))|Xy with graded pieces Ns Ns (Ωs (Xy ) ⊗ Λq−s+1 (M )) (F s /F s−1 ) = ⊗s=q ⊗s=q s=0 Sym s=0 Sym

for all multi-indices (N0 , . . . , Nq ) of sum N . The terms of the (multi)ﬁltration are indexed by N -tuples (s1 , . . . , sN ) of integers 0 ≤ sh ≤ q, corresponding to tensors u1 ⊗· · ·⊗uN of lexicographically ordered ≤(q−sh ) h ∗ local sections uh ∈ (Ω≥s ), for each h = 1, . . . , N . X ) ∧ f (ΩY By taking the saturations of the pieces of this ﬁltration into S N (Ωq (X|Δ)), and restricting to a generic Xy , we get a ﬁltration on S N (Ωq (X|Δ))|Xy with graded Cs=q pieces of the form s=1 S Nh (Ωh (Xy |Δy )) ⊗ Mh , in which the Mh are trivial bundles on Xy , tensor products of symmetric products of the conormal bundle of Xy in X. We describe this ﬁltration locally now. In local adapted coordinates (z) = (z1 , . . . , zn ), with Δ supported on the coordinate hyperplanes zj , j = 1, . . . , (n − p), and f (z) = (y1 := zn−p+1 , . . . , yp := zn ), recall that a basis of the OX -module S N (Ωq (X|Δ)) is given by the following: dz(J) 1 := ⊗\=N dz(JE ) , z(J) z(J) \=1

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for (J) = (J1 , . . . , JN ) any (lexicographically ordered) N -tuple of subsets J\ of {1, 2, . . . , n} of cardinality q, where j=(n−p)

z(J) := Πj=1

3kj (1−1/mj )@

(zj

),

and kj is the number of ]’s in {1, . . . , N } such that j ∈ J\ . Now each subset J\ can be uniquely written as a disjoint union J\ = H\ ∪ K\ , with H\ ⊂ H := {1, 2, . . . , (n − p)}, and K\ ⊂ K := {(n − p + 1), . . . , n}. Thus r=q dz(J) := ⊗\=N \=1 dz(JE ) = ⊗r=0 dz(J) (r), where dz(J) (r) := ⊗{\|r(\)=r} dz(JE ) , and, for each ], r(]) is deﬁned to be the cardinality of H\ . This expression is the local splitting of the ﬁltration of SymN (Ωq (X))|Xy described above. This ﬁltration naturally induces a corresponding ﬁltration on S N (Ωq (X|Δ))|Xy , with graded pieces locally given in the coordinates above by dz(J) dz(J) (r) 1 := ⊗r=q dz(J) (r) = ⊗r=q , r=0 z(J) z(J) r=0 z(J) (r) where, as above j=(n−p)

z(J) := Πj=1 while

j=(n−p)

z(J) (r) := Πj=1

3kj (1− m1 )@

(zj

j

),

3kj (r) (1− m1 )@

(zj

j

),

the integers kj (r) being deﬁned as follows: kj (r) is the number of ]’s in {1, . . . , N } such that j ∈ J\ and r(]) = r, for r = 0, . . . , q. From the deﬁnitions and the equality above it follows that, for any (J): " %r=q j=(n−p) kj z(J) (0) := Πj=1 (zj ), with kjD := \kj (1 − 1/mj )L − r=1 \kj (r)(1 − 1/mj )L. The following trivial estimate will be crucial, here: 4.24. Lemma. We have \kj (0) (1 − 1/mj )L ≤ kjD ≤ q + \kj (0) (1 − 1/mj )L, for any N, n, p and j ∈ {1, . . . , (n − p). Proof. This follows from the fact (applied to xr := kj (r) (1 − 1/mj )), that the integral part of the sum of q non-negative real numbers xr , r = 1, . . . , q, lies between the sum of their integral parts, and this same sum increased by q. Step Two. The second step of the proof of 4.23 consists in showing that any nonzero section s of S N (Ωq (X|Δ)), when restricted to Xy , has its image contained in the smallest piece of the ﬁltration above, described as dz(J) (0)/z(J) (0) in the local coordinates above. But this follows immediately from the hypothesis made, and the ﬁltration given, since otherwise the graded pieces dz(J) (r)/z(J) (r), for r > Ns (Ωs (Xy |Δy )) ⊗ 0, would induce non-zero sections over Xy of some sheaf ⊗s=q s=0 S h Λ (M ), which are supposed not to exist. Step Three. The third step will show now that the given section is of the form f ∗ (w) for some local section w of SymN (ΩqY ) near y ∈ Y , generic. From the description of the last term dz(J) (0)/z(J) (0) of the ﬁltration above, we see from Lemma 4.24 that s is, near y ∈ Y , a section of f ∗ (SymN (ΩqY ))(q D), where D is

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the support of Δ. Notice now that the ﬁltration above is compatible with tensor products. Thus replacing s with a suﬃciently high tensor power s⊗k , the (obvious) lemma 4.25 below shows that s has, in fact, no poles along D. Thus s = f ∗ (w) for some section w of f ∗ (SymN (ΩqY )) near generic y in Y . 4.25. Lemma. Let s be a meromorphic section of a line bundle L on the complex manifold X. Let D be a reduced divisor on X. Assume that, for some given integers k > q > 0, the meromorphic section s⊗k has poles of order at most q along D, and is holomorphic outside of D. Then s is holomorphic. Step Four. The fourth and last step consists in showing that the (generically uniquely) deﬁned section w of SymN (ΩqY ) extends meromorphically to Y as a section of S N (Ωq (Y |ΔY )), where ΔY := Δ(f |Δ) . Because the sheaf SymN (ΩqY ) is locally free, it is suﬃcient, by Hartog’s theorem, to show this extension at the generic point of any prime divisor E on Y . If f is submersive over the generic point y of E, and no component of D, the support of Δ, is mapped onto E, this is clear, since one can take a local section g of f over a neighborhood U of y in Y ; restrict s to the image of this section. Then g ∗ (s) and w coincide at the generic point of U , and so g ∗ (s) is the desired extension of w. We shall adapt this easy argument to the orbifold context. Let indeed mE be the multiplicity of E in the orbifold ΔY . There is an irreducible divisor F ⊂ X such that mE = t mF , where mF is the multiplicity of F in Δ = ΔX , and t is the positive integer such that f ∗ (E) = t F + . . . , that is the multiplicity of F in f ∗ (E). Assume ﬁrst that t = 1. Then we have mE = mF , and moreover a local section g of f exists. The very same argument as above applies, and thus gives the desired extension g ∗ (s) of w, but this time g ∗ (s) is only a section of SymN (Ωq (Y |ΔY )), since s is a section of the restriction to g(U ) of SymN (Ωq (X|Δ)), and g : (Y |ΔY )|U → (X|Δ)|g(U ) is a local isomorphism of orbifolds, by suﬃciently shrinking U , and choosing g(y) suﬃciently generic in Fy . The general case where t > 1 is now deduced from the case t = 1 by making a local base change over h : U D → U which ramiﬁes at order t along E. One gets by the preceding argument an extension σ D := (g D )∗ (sD ) of wD := h∗ (w) to U D , as a section of SymN (Ωq (Y D |ΔY " )) over U D , where ΔY " is deﬁned over U D as the single divisor E D = h−1 (E), equipped with the multiplicity mE /t. A simple local computation (in dimension 1, in fact) shows that σ D = h∗ (σ), for σ a section of SymN (Ωq (Y |ΔY )) over U . Indeed, the potential poles of σ are contained in E, with multiplicities at most t (coming from a diﬀerence in round-downs), independent of N . Thus Lemma 4.25 above applies again, by replacing s with s⊗(t+1) .

5. Orbifold Additivity The orbifold version of Iitaka’s Cn,m -conjecture is: orb 5.1. Conjecture. (Cn,m ) Let (f |Δ) : (X|Δ) → Z be a ﬁbration, with (X|Δ) smooth and X projective. Then: κ(X|Δ) ≥ κ(Yz |Δz ) + κ(f |Δ). Here Δz is simply the restriction of Δ to Xz := f −1 (z) with z ∈ Z generic.

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Our main technical result is the following. orb 5.2. Theorem. The Cn,m -conjecture is true if κ(f |Δ) = dim Z (i.e. if (f |Δ) is of general type). In this case one has κ(X|Δ) = κ(Xz |Δz ) + dim Z.

5.3. Corollary. Let (X|Δ) be a smooth projective orbifold, and let f : X → Y and g : Y → Z be ﬁbrations. Then: κ(f |Δ) = κ(fz |Δz ) + dim Z, if g ◦ f is of general type, denoting (fz |Δz ) : (Xz |Δz ) → Yz the restriction over a general z ∈ Z. Thus: if (g ◦ f |Δ) and (fz |Δz ) are of general type, then (f |Δ) is of general type. This is a fairly direct application of the following semi-positivity result ([Ca04, Theorem 4.11, p. 567], diﬀerently formulated), extending [Vieh 83]: 5.4. Theorem. [Ca 04, Theorem 4.11, p. 567] Let (X|Δ) be a smooth geometric, with X projective. Let f : X → Z be a ﬁbration with Z smooth. Assume that f is ‘prepared’ (see [Ca 04, 1.1.3, p. 508])15 . Let (Z|ΔZ ), with ΔZ := Δ(f |Δ) be the orbifold base of (f |Δ). Deﬁne KX/(Z|ΔZ ) := KX − f ∗ (KZ + ΔZ ). Then, f∗ (N · (KX/(Z|ΔZ ) + Δ)) is weakly positive in Viehweg’s sense for any N > 0 such that N · (KX/(Z|ΔZ ) + Δ) is Cartier. In particular, the restriction of F := m (N · (KX/(Z|ΔZ ) + Δ)) + f ∗ (B) over the generic ﬁber Xy of f is generated by the global sections of F on X for any given N as above and any big Q-divisor B on Y if m is suﬃciently large (depending on N, B). A result of a similar nature, but more precise (the conclusion is nefness), is shown in [Ka 98], when the orbifold ﬁbres have a trivial canonical bundle. The proofs are essentially adaptations to the orbifold context of Viehweg’s proof of the Cn,m -conjecture for ﬁbrations with base a variety of general type ([Vieh 83]). We shall not give the rather technical and lengthy details here. The range of applications is however considerably increased by the orbifold context. It is, for example, one of the crucial ingredients used in [HM 05] and [HM 05b]. A diﬀerent proof is given in [BP 07]. 5.5. Corollary. Let (f |Δ) : (X|Δ) → Z be a ﬁbration of general type, then: κ(X|Δ) ≥ κ(Xz |Δz ) + dim Z. 5.6. Corollary. Let (X|Δ) be a smooth projective orbifold with κ(X|Δ) = 0. There is no ﬁbration f : X ''( Z of general type on (X|Δ). 5.7. Example. Let (X|Δ) be the smooth projective orbifold with X a projective toric manifold, and Δ the reduced anticanonical divisor complement of the open orbit. There is no ﬁbration f : X ''( Z of general type on (X|Δ). (Example suggested by a question of M. Mustat¸a). 5.8. Corollary. Let (X|Δ) be a smooth projective Fano orbifold (i.e. such that −(KX + Δ) = H is ample on X). There is no ﬁbration f : X ''( Z of general type on (X|Δ). 15 This

can be achieved by an elementary orbifold modiﬁcation of (X|Δ), which does not change the space of sections of N · (KX + Δ). The term is introduced in [Vieh 83].

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Proof: Choose a reduced D ∈ |N H|, for N large and divisible such that ΔD := 1 D N D+Δ has a support of normal crossings. Then (X|Δ ) is smooth, and has trivial canonical bundle. Since there is, by 5.6, no ﬁbration of general type on (X|ΔD ), there is, a fortiori, no ﬁbration of general type on (X|Δ). orb 5.9. Remark. Conjecture Cn,m implies much more: if (f |Δ) : (X|Δ) ''( Z is a ﬁbration from (X|Δ), smooth and projective, it should follow that: 1) κ(f |Δ) ≤ 0 if κ(X|Δ) = 0. 2) κ(f |Δ) = −∞ if κ(X|Δ) = −∞ and κ(Xz |Δz ) ≥ 0.

In fact, even ‘more’ should be true. This ‘more’ is best formulated with the obvious orbifold versions of κ++ and κ+ : 5.10. Deﬁnition. Let (X|Δ) be a smooth projective complex connected orbifold. Let: κ++ (X|Δ) := max{L,p>0} {κ(X|Δ, L)}, where L ⊂ ΩpX is a rank 1 coherent subsheaf. (See Deﬁnition 4.13 for the deﬁnition of κ(X|Δ, L)). Deﬁne also κ+ (X|Δ) := max{f } {κ(f |Δ)}, where (f |Δ) : (X|Δ) ''( Z ranges over all ﬁbrations deﬁned on X. Thus: dim(X) ≥ κ++ (X|Δ) ≥ κ+ (X|Δ) ≥ κ(X|Δ) for any (X|Δ). 5.11. Remark. 1. Notice that, when Δ = 0, it is not obvious that the old κ+ (X) of 2.3 coincides with the new κ+ (X|0) in the sense of 5.10, which is greater. But Conjectures 5.1 and 2.4 imply that κ+ (X|0) = κ+ (X). 2. Conjecture 9.10 in §9 asserts that, just as when Δ = 0, the condition κ+ = −∞ is equivalent to orbifold rational connectedness. 5.12. Conjecture. If κ(X|Δ) ≥ 0 then κ++ (X|Δ) = κ(X|Δ). Moreover, if κ+ (X|Δ) = −∞ then κ++ (X|Δ) = −∞. orb and 5.12 will be deduced from standard conjectures of the The conjectures Cn,m LMMP in §10. The central conjecture (see 11.7) concerning families of canonically orb polarized manifolds will be shown to follow from Conjectures Cn,m and 5.12 in §11.D, and thus from standard conjectures of the LMMP.

6. Special Orbifolds 6.A. Deﬁnition and main examples. 6.1. Deﬁnition. A smooth projective orbifold (X|Δ) is ‘special’ if there does not exist a dominant rational map (f |Δ) : (X|Δ) ''( Y of base-general type (i.e. with κ(f |Δ) = dim Y > 0). A variety X is special if so is (X D |0), for some (or any) smooth model X D of X. A quasi-projective manifold U is special if so is (X|D) for some (or any) smooth compactiﬁcation X of U by a normal crossing divisor D. Remarks. 0. Being special is indeed a birational property. 1. A (smooth projective, as always in the sequel) orbifold (X|Δ) is special if and only if there does not exist a Bogomolov sheaf on (X|Δ) (by Theorem 4.15).

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2. An orbifold of general type and positive dimension, is not special (consider the identity map). 3. An orbifold curve (X|Δ) is special if and only if deg(KX + Δ) ≤ 0, since a ﬁbration f : X ''( Y is either the identity map or the constant map. 4. A rationally connected manifold X is special, since κ++ (X) = −∞. 5. A Fano orbifold is special, by Corollary 5.8. 6. Any orbifold with κ = 0 is special, by Corollary 5.6. 7. Any orbifold with κ+ = −∞ is obviously special. 8. The previous two examples are in fact the ‘building blocks’ of special orbifolds. First, Theorem 6.2 shows that towers of ﬁbrations with general orbifold ﬁbres having either κ = 0, or κ+ = −∞ are special. Conversely, we shall see in orb , any special orbifold decomposes canonically section 8 that, conditionally on Cn,m as a tower of such ﬁbrations. 6.2. Theorem. Let (f |Δ) : (X|Δ) → Y be a neat ﬁbration, with (X|Δ) smooth projective. Assume that its general orbifold ﬁbre (Xy |Δy ) and orbifold base (Y |Δ(f |Δ)) are special. Then (X|Δ) is special. The proof is sketched in 8.7. 6.3. Remark. 1. This statement is (very) false if one does not take into account the orbifold structures on both ﬁbres and base (see Example 2.12 when Δ = 0). 2. Orbifolds with either κ = 0 or κ+ = −∞ are special, while manifolds of general type are not special, so one might ask if further relations between κ and specialness hold. This is not the case: for every n ∈ N and every k ∈ {−∞, 1, . . . , n − 1}, there exist special manifolds X of dimension n and κ = k. For example, any hyperplane section X ⊂ Pn+1−k × Pk of bidegree (n + 2 − k, d), with d > k + 1, and containing a section over Pk , has dim X = n, κ(X) = k if k > 0, and is special. 6.B. Criteria for special orbifolds. 6.4. Theorem. A smooth projective orbifold is special if (and only if ) two generic points are joined by a chain of special suborbifolds (i.e. images of orbifold morphisms from special smooth orbifolds). The proof is sketched in the next section 7. See Remark 7.4. 6.5. Remark. When Δ = 0, 6.4 does not generalize to arbitrary singular varieties (e.g. if X is a cone over a variety of general type Y ). It should however hold for varieties with log-canonical singularities. A special manifold does not necessarily admit non-trivial chains of special subvarieties (e.g. if X is a simple abelian variety). 6.6. Theorem. Let g : X D ''( X be a dominant map. If X D is special, so is X. If X is special, and if g is regular and ´etale, then X D is special. The proof of the seemingly easy second assertion requires the diﬃcult result 5.2.

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6.7. Theorem. Let ϕ : Cn → X be a non-degenerate meromorphic (possibly transcendental) map. Then X is special. This extends a former result of Kobayashi-Ochiai asserting that κ(X) < dim X under the hypothesis of non-degeneracy above (which means that ϕ is holomorphic and submersive at some point). More general versions of 6.7 are given in [Ca 04]. 6.C. Special surfaces. They can be easily described, using classiﬁcation. Such a simple description however fails in higher dimensions. 6.8. Proposition. The special surfaces X are exactly the following ones (up to birational equivalence): κ(X) = −∞: then X @ P1 × C with g(C) = 0 or 1. κ(X) = 0: then X is a K3 surface or abelian. κ(X) = 1: (after a suitable ´etale cover of X) the Moishezon-Iitaka elliptic ﬁbration X → C has either g(C) = 1 and no multiple ﬁbre, or g(C) = 0, and at most 2 multiple ﬁbres of coprime multiplicities. 6.9. Deﬁnition. A manifold X is ‘weakly special’ if no ﬁnite ´etale cover X D → X has a meromorphic map f : X D ''( Y onto a variety Y of general type with dim Y > 0. It follows from Theorem 6.6 that if X is special, it is weakly special, the converse being true when dim X ≤ 2. Indeed: 6.10. Corollary. Let X be a smooth surface. The following are then equivalent: 1) X is special. 2) X is ‘weakly special’. 3) κ(X) ≤ 1, and π1 (X) is almost abelian. 4) There exists a nondegenerate map h : C2 ''( X. 6.11. Remark. 1) None of these characterizations of specialness extends to higher dimensions. 2) The property of being special is thus invariant by deformation for surfaces. 3) There exist weakly special threefolds X which are not special. See Example 12.2 for a sketch of their construction). These threefolds show that orbifolds are needed in the birational classiﬁcation theory of projective varieties (multiple ﬁbres cannot always be eliminated by ﬁnite ´etale covers), and also permit to test some conjectures in arithmetics and hyperbolicity (see section 11.A below).

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7. The Core The main structure result of this text is: 7.1. Theorem. For any smooth complex projective orbifold (X|Δ), there exists a unique map c = c(X|Δ) : X ''( C(X|Δ), called ‘the core’ of (X|Δ) such that: 1) The ‘general’ orbifold ﬁbre (Xc |Δc ) of c is special. 2) κ(c|Δ) = dim C(X|Δ) ≥ 0. Moreover, c is almost holomorphic (by Theorem 4.16). 7.2. Remark. We do not know if the orbifold bases of ‘neat’ models of c are birational invariants. Idea of proof: Uniqueness follows from Lemma 7.3 below. For the existence: Let f : X ''( Y be a ﬁbration with d := dim Y maximal such that κ((f |Δ)) = dim Y . Thus d = 0 (resp. d = dim X) if and only if (X|Δ) is special (resp. of general type). In the general case, we have to show that the ﬁbres of (f |Δ) are special. Assume not. Then by Chow space theory and countable upper semi-continuity of the dimension d above, one can construct a factorisation f = g ◦ h, where: h : X ''( Z, g : Z → Y with dim Z > dim Y and such that for general y ∈ Y , the map: (hy |Δy ) : (Xy |Δy ) → Zy is of general type. But it then follows from Corollary 5.3 that (h|Δ) : (X|Δ) → Z is a ﬁbration of general type, contradicting the maximality of dim Y . 7.3. Lemma. Let (h|Δ) : (X|Δ) ''( Z be a ﬁbration with general orbifold ﬁbre special (resp. let (g|Δ) : (X|Δ) ''( Y be of general type). Then there exists a unique map u : Z → Y such that g = u ◦ h. Idea of proof: Otherwise the images of the general ﬁbres Xz of h by g are positive-dimensional. But one can show that the restriction of the corresponding Bogomolov sheaf L(g|Δ) on (X|Δ) to such an orbifold ﬁbre (Xz |Δz ) would then be a non-zero Bogomolov sheaf on (Xz |Δz ) which is special. This is a contradiction. 7.4. Remark. The map c enjoys the following properties (as may be seen from the sketch of proof of 7.1 above): 1. dim C(X|Δ) = n (resp. 0) iﬀ (X|Δ) is of general type (resp. special). 2. If V ⊂ X is a subvariety which meets a (suitable) general ﬁbre Xc of cX , and is the image of an orbifold morphism (W |ΔW ) → (X|Δ), with (W |ΔW ) special, then V ⊂ Xc . This property implies Theorem 6.4. 3. If u : (X D |ΔD ) → X is a surjective orbifold morphism, there exists a unique map cu : C(X D |ΔD ) → C(X|Δ) such that c(X|Δ) ◦ u = cu ◦ c(X " |Δ" ) . 4. If u is ﬁnite and orbifold-´etale, cu is generically ﬁnite. 5. If (f |Δ) : (X|Δ) ''( Y is a ﬁbration with general orbifold ﬁbre special (resp. if (g|Δ) : (X|Δ) ''( Z is of general type), there exist unique maps u : Y → C(X|Δ) (resp. v : C(X|Δ) ''( Z) such that c(X|Δ) = u ◦ f (resp. g = v ◦ c(X|Δ) ). Use Lemma 7.3. The ﬁrst case applies for example to the ‘rational quotient’ r of (X|Δ) (see 8.3 below) and to its Moishezon-Iitaka ﬁbration M if κ(X|Δ) ≥ 0 (see 8.1 below).

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7.5. Example. 1. Let MX : X → P1 be the Moishezon-Iitaka ﬁbration of Example 2.12, in the notation of §2.F. This is the core of X. 2. Let MX : X → M (X) be the Moishezon-Iitaka ﬁbration of any subvariety of an Abelian variety. This is the core of X, by [U 75], Thm. 10.3, p. 120. 3. Using the last property of Remark 7.4 above, one sees that for any X, the composite map M ◦ r : X ''( M R(X) (well-deﬁned if κ(R(X)) ≥ 0, which conjecturally always holds) has special ﬁbres, since Δ(r) = 0, by [GHS 03]. Thus M ◦ r factorises cX (i.e. cX = u ◦ (M ◦ r) for some u : M R(X) ''( C(X)). We will now generalise this factorisation.

8. The Decomposition of the Core Even if we start with some X without orbifold structure (i.e. Δ = 0), we are, in general, immediately faced with the existence of nonzero Δ’s in the orbifold bases of the decomposition process (M ◦ r)n described above (in §2.F), which needs to be run in the orbifold category. So we state the decomposition structure in this latter category. We need to deﬁne the maps r and M in this orbifold context. We always denote by (X|Δ) a smooth projective orbifold. The deﬁnition of M for orbifolds does not present any new diﬃculty: 8.1. Proposition. Assume that κ(X|Δ) ≥ 0. Then there exists a (birationally) unique ﬁbration M = M(X|Δ) : (X|Δ) → M (X|Δ) such that 1) Its orbifold ﬁbres have κ = 0. 2) dim(M(X|Δ) ) = κ(X|Δ). The construction is the same as when Δ = 0. 8.2. Remark. We do not know if the generic orbifold ﬁbres or orbifold base of ‘neat’ models of M are birational invariants. 8.A. Weak orbifold rational quotient. We shall now deﬁne an orbifold weak orb version of the ‘rational quotient’ 2.5 in the orbifold context, assuming Cn,m . 8.3. Theorem. Let (X|Δ) be a smooth geometric orbifold. There exists a (birationally) unique map r = r(X|Δ) : X ''( R = R(X|Δ) such that orb 1) κ(r|Δ) ≥ 0 (assuming Cn,m ). 2) Its generic orbifold ﬁbres (Xs |Δs ) satisfy: κ+ (Xs |Δs ) = −∞. (The generic orbifold ﬁbres are well-deﬁned since r is almost holomorphic, by the ﬁrst property). The map r is called the ‘κ-rational quotient’ of (X|Δ). Idea of proof: Take a dominant connected map f : X ''( Y with dim Y maximal such that κ(f |Δ) ≥ 0. If the generic orbifold ﬁbre doesn’t have κ+ = −∞, one can construct a new map g : X ''( Z with dim Z > dim Y such that κ(g|Δ) ≥ 0, orb using Cn,m .

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8.4. Remark. 1) For any smooth orbifold (X|Δ), the composite map M ◦ r is orb always deﬁned, since κ(r|Δ) ≥ 0 (the existence part assumes Cn,m ). 2) M ◦ r is the identity map of X if and only if (X|Δ) is of general type. 3) We do not know if the orbifold bases of ‘neat’ models of r are birational invariants. 4) The map r is almost holomorphic if (X|Δ) is ﬁnite, but not necessarily if (X|Δ) is logarithmic with lc, but not klt singularities. See Example 4.17. 8.5. Example. Consider the special case where X is a smooth surface, and Δ = 0: r is the identity map if and only if κ(X) ≥ 0, and has image a curve B if and only if κ(X) = −∞ and q(X) > 0, it is the constant map if and only if X is rational. The map M ◦ r is the identity map if and only if κ(X) = 2, it maps to a curve if and only if either κ(X) = 1, or κ(X) = −∞ and q(X) ≥ 2. Next, M ◦ r is the constant map if and only if either κ(X) = 0, or κ(X) = −∞, and q(X) ≤ 1. Thus (M ◦ r) = (M ◦ r)2 is the core of X, except when κ(X) = 1 and the orbifold base (B|Δ(M )) of M : X → B is not of general type. We then have either: κ(B|Δ(M )) = −∞, or: κ(B|Δ(M )) = 0. In the ﬁrst case, r ◦ M ◦ r is the constant map. In the second case, (M ◦ r)2 is the constant map. In both cases X is special, and (M ◦ r)2 = c, the core of X. This example generalises as follows. 8.B. The conditional decomposition c = (M ◦ r)n . orb . For any smooth projective n-dimensional (X|Δ), 8.6. Theorem. Assume Cn,m n one has: c = (M ◦ r) . Here c is the core, r the κ-rational quotient, and M the orbifold Moishezon-Iitaka ﬁbration, respectively deﬁned in 8.3 and 8.1.

Idea of proof: After Remark 8.4, for any k ≥ 0 and any smooth (X|Δ), the map (M ◦ r)k can be deﬁned. Moreover, if (M ◦ r)k+1 = (J ◦ r)k (which holds for k = n, by reasons of dimension), then (M ◦ r)k is a ﬁbration of general type. Thus (M ◦ r)n is a ﬁbration of general type. By Theorem 8.7 below, it has also (by induction on k) special orbifold ﬁbres, and the asserted equality is established (by uniqueness of c). 8.7. Theorem. Let (X|Δ) be smooth, and let f : X → Y be a neat ﬁbration. If the orbifold base and the general orbifold ﬁbre of f are special, then (X|Δ) is special. 8.8. Remark. This result is very false in the category of manifolds (just considering ﬁbres and usual base, as seen in Example 2.12). It shows one of the essential improvements of working in the orbifold category. Another similar ‘additivity’ result is the exact sequence ([Ca 07], §12) of orbifold fundamental groups for ﬁbrations. Idea of proof (of 8.7): Assume not. Let c : (X|Δ) → C be the core, assuming dim(C) > 0. The orbifold ﬁbres of f being special are contained in those of c (which are maximal for this property). Hence there exists a factorisation c = g ◦ f , for some g : Y → C. On suitable birational models, this induces an orbifold

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morphism: g : (Y |ΔY ) → (C|Δ(c|Δ) ). This contradicts the specialness of (Y |Δ) and the hypothesis: κ(C|Δ(c|Δ) ) = dim(C) > 0. We can now get (conditionally) a more concrete description of special orbifolds: orb 8.9. Corollary. Assume Cn,m . A smooth projective n-dimensional (X|Δ) is n special if and only if (M ◦ r) is the constant map.

8.10. Remark. Thus special orbifolds are exactly the ‘orbifold combinations’ of orbifolds with either κ = 0, or κ+ = −∞ (i.e. are towers of ﬁbrations with orbifold ﬁbres having either κ = 0, or κ+ = −∞). Even when Δ = 0, the consideration of orbifold structures is essential, as shown again by Example 2.12.

9. Orbifold Rational Curves The objective of these notions is to formulate Conjecture 9.10 below, asserting, just as when Δ = 0, the equivalence between the condition κ+ = −∞ and orbifold rational connectedness. The notion of rational curve in the orbifold context is more involved than for manifolds, and their geometry seems to be more diﬃcult to study, too. See [Ca 07, Chap. 5] for more details. We shall deﬁne the notions of uniruledness and rational connectedness for smooth geometric orbifolds (X|Δ). The objective being to establish for these orbifold rational rational curves the same results ([MiMo 86], [Mi 93], [KMM 92], [KMM 92b]) as when Δ = 0. We shall here consider mainly the ‘divisible’ orbifold rational curves, and shall give a very brief survey of the problems arising. See again [Ca 07], §5 for more details. 9.A. Minimal orbifold divisors. Let (X|Δ) be a smooth integral projective orbifold, and let g : C → (X|Δ) be a morphism from a connected % smooth projective curve C such that g(C) 2 Supp(Δ). Write (as usual): Δ := j (1 − 1/mj ) Dj . There thus exists a unique smallest integral orbifold divisor ΔC on C such that g : (C|ΔC ) → (X|Δ) is a ‘divisible’ orbifold morphism16 . Explicitly, for any point a ∈ C, the ΔC -multiplicity mC (a) of the divisor {a} on C is given by: • mC (a) = 1 if f (a) ∈ / Supp(Δ). m • mC (a) = lcmj∈J(a) { gcd(mjj,tj,a ) }, where J(a) := {j ∈ J such that g(a) ∈ Dj } and g ∗ (Dj ) = tj,a {a} + . . . with tj,a being the order of contact of g∗ (C) with Dj at g(a). (By convention, the only multiple of +∞ is itself.) 9.1. Example. 1. mC (a) = 1 iﬀ for each j ∈ J(a) we have that mj divides tj,a . 2. If Δ is logarithmic, ΔC = 0 iﬀ g(C) is disjoint from Δ. 16 One can deﬁne orbifold Q-rational curves (or curves which are ΔQ -rational) similarly. In this mj case, mQ }, instead. C (a) = maxj∈J(a) { t j,a

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9.B. Orbifold rational curves. 9.2. Deﬁnition. A (‘divisible’)17 rational curve on (X|Δ) (or: a Δ-rational curve on X) is a map g : C → (X|Δ) such that κ(C|ΔC ) = −∞. (This implies that C∼ = P1 , and Supp(ΔC ) has at most 3 points with multiplicities at most (2, 3, 5)). When R ⊂ X is a rational curve not contained in Supp(Δ), we say that R is ˆ→ a Δ-rational curve if so is, in the sense above, the ‘normalised’ inclusion g : R (X|Δ) obtained by composing the inclusion of R in X with the normalisation18 of R. 9.3. Example. 1. If R, rational, has all its orders of contact with each Dj divisible by mj , then R is Δ-rational with ΔC = 0. 2. If (X|Δ) is smooth logarithmic (Δ = supp(Δ)), the Δ-rational curves are the rational curves R on X whose normalisation meets Δ in at most 1 point. For example, if X = P2 , and Δ = D is a projective line with inﬁnite multiplicity, a line L (resp. an irreducible conic C, resp. an irreducible singular cubic Q) is Δrational if and only if L K= D (resp. C is tangent to D, resp. Q is cuspidal, and tangent to D in its unique singular point). 3. If X = P2 , and if Δ is the union of 3 lines in general position, with arbitrary integral multiplicities a, b, c, then (P2 |Δ) is Fano. Indeed: 3 − [(1 − a1 ) + (1 − 1b ) + (1 − 1c )] < 0. There are three families of Δ-rational lines covering X, those passing through one of the three intersection points of any two of the three given lines. We shall see in 9.13 and 9.17 that, for any ﬁnite subset E of P2 , there exists a curve, Δ-rational irreducible, containing E. 4. Let X = P2 , and Δ be the union of 4 lines in general position, with multiplicities 2, 2, a, b, for arbitrary integers 2 ≤ a ≤ b. Thus (P2 |Δ) is Fano, since 1 1 1 1 1 1 3 − [(1 − ) + (1 − ) + (1 − ) + (1 − )] = 2 − [(1 − ) + (1 − )] < 0. 2 2 a b a b In this case, if a ≥ 4, only one of the three preceding families of curves consists of Δ-rational curves: those passing through the intersection of the two lines of multiplicities a and b. If we replace the two lines of multiplicity 2 by a conic of multiplicity 2, we get a second family of lines which are Δ-rational: the family of tangents to C. 5. Consider now X = P2 , and Δ the union of 4 lines in general position, with multiplicities 3, 3, 5, 7. Thus (P2 |Δ) is Fano, since 1/3 + 1/3 + 1/5 + 1/7 = 106/105 > 1. The lines which are ΔQ -rational are ﬁnite in number, since a line meeting Supp(Δ) in at least 3 points, gets multiplicities at least 3, 3, 7, and 1/3 + 1/3 + 1/7 < 1. It does not seem obvious to produce an explicit covering family of Δ-rational curves in this case. A dimension count however shows that this might be possible (even with ΔD = 0), but only for large degrees, divisible by 3 · 5 · 7 = 105. Indeed, Q-rational curves are deﬁned similarly, using the orbifold divisor ΔQ C , instead. [Ca 07, §5], maps g : R → (X|Δ) which are not necessarily birational to their image are considered, too. They should be important to obtain more deformations. 17 Orbifold

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rational irreducible plane curves of degree d = N · 105 depend on pN := 3(d + 1) − 1 − 3 = 3d − 1 parameters. Having with a line only points of contact of orders all divisible by dD , divisor of d, depends on dd" (dD − 1) = d (1 − d1" ) conditions. Thus, in our case, we get (with dD = 3, 3, 5, 7 successively), a number of conditions in total equal to 1 1 1 1 1 ) = 3d − 3N. cN := d [(1 − ) + (1 − ) + (1 − ) + (1 − )] = d (3 − 3 3 5 7 105 We can thus expect to have a family of such curves depending on: pN − cN = 3N − 1 parameters of Δ-rational curves (without orbifold structure, even). See also Example 9.11 below. 6. The preceding example can be considered with multiplicities (a ≤ b ≤ c ≤ d) instead of (3, 3, 5, 7). The degree of the canonical bundle is then δ := −3 + [(1 − a1 ) + (1 − 1b ) + (1 − 1c ) + (1 − d1 )], so that (P2 |Δ) is Fano if and only if −δ = a1 + 1b + 1b + d1 − 1 > 0; for example if (a, b, c, d) = (2, 3, 7, 41). Notice that, in this last case, the orbifold considered is simply connected, and has, in particular, no orbifold ´etale cover. The dimension count above still applies in this context and shows that rational curves of degree d = N m, where m is divisible by lcm(a, b, c, d), should depend on, at least, dm(−δ) − 1 parameters. Observe that d(−δ) − 1 = −(KP2 + Δ)R + 2 − 3 if R is a curve of degree dm. The right-hand side of this last equality is, in fact, the Euler characteristic of the lift of the orbifold tangent sheaf to R, which computes the obstructions to deforming a orbifold morphism P1 → (X|Δ) with direct image R. See [Ca 07], Remark 5.8 for more details. 7. Let X be a smooth toric projective manifold. Let D be its toric anticanonical divisor (it has normal crossings). Equip each of the components of D with a ﬁnite multiplicity, and let Δ be the resulting orbifold divisor. If R is a rational toric curve (closure of the orbit of a one-parameter algebraic subgroup of the torus acting on X), not contained in D, then R meets D in at most 2 points in which it is unibranch. It is thus a Δ-rational curve. 8. By contrast, consider the logarithmic smooth orbifolds (P2 |Δ), Δ := L + LD and (P2 |ΔD ), ΔD := C, where L, LD are two distinct lines, and C is a smooth conic. Both are Fano. The only Δ-rational curves are, however, the lines through the intersection point (L ∩ LD ), while there exists, for every d > 0, an explicit (d + 1)dimensional family of ΔD -rational curves of degree d. The reason for this diﬀerent behaviour lies in the lc, but not klt singularity of (P2 |Δ). See [Ca 07], Example 5.14. 9.C. Orbifold uniruledness and rational connectedness. 9.4. Deﬁnition. A smooth (X|Δ) is uniruled (resp. R.C.) if any generic x (resp. (x, y)) in X is contained in a Δ-rational curve. 9.5. Example. 1) Let (X|Δ) be the toric example 9.3.7 above. Then (X|Δ) is rationally connected, since all of its toric rational curves (those not contained in D) are Δ-rational. 2) The orbifolds (P2 |Δ) of Examples 9.3.3, 4 are RC, by the

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explicit covering families of Δ-rational curves displayed there. 3) The orbifolds (P2 |Δ) of Examples 9.3.5, 6 should be RC, by the counting argument given there. Some properties of rationally connected manifolds extend immediately: 9.6. Proposition. Assume (X|Δ) is rationally connected. Then: 1) π1 (X|Δ) is ﬁnite. 2) H 0 (X, S N (Ωp (X|Δ))) = 0, for any N, p > 0. Thus κ++ (X|Δ) = −∞. See [Ca 07] for a proof. 9.7. Remark. It is not obvious (if true) that orbifold uniruledness and rational connectedness are birationally invariant properties (for ﬁnite, integral, smooth) orbifolds. Indeed, some orbifold rational curves may not lift under an (orbifold) blow-up. An example is given in [Ca 07, 5.8]. 9.D. Fano orbifolds. The following question is the easiest one of a series of questions asking whether results known when Δ = 0 extend to the orbifold context: 9.8. Question. Let (X|Δ) be a smooth, integral, ﬁnite geometric orbifold, with X projective. Assume (X|Δ) is Fano (i.e. −(KX + Δ) is ample). Is then (X|Δ) uniruled (resp. rationally connected)? Is this at least true when Pic(X) ∼ = Z? 9.9. Example. Assume that (X|Δ) is smooth, ﬁnite and Fano, with X projective and Pic(X) ∼ = Z generated by the ample line bundle H. Let r ≤ (n + 1) be the divisibility index19 of X. Assume that Δ = (1 − 1/m) · D, with D a member of |kH| with only normal crossings as singularities. Thus (1 − 1/m)k < r, since (X|Δ) is supposed to be Fano. Assume, more strongly, that k ≤ r. It follows then from [KM 98, Theorem 5.8, p. 28] that there exists a covering family of rational curves on X meeting D in at most two points (after normalisation). In other words: (X|D) is uniruled. Thus so is (X|Δ). The assumption that k ≤ r is met in particular when r = 1. The situation thus seems to be more involved when r increases, in particular when it is maximal, equal to (n + 1), that is, when X ∼ = Pn . Other questions ask whether Miyaoka-Mori’s Bend and Break, Miyaoka’s generic semi-positivity, Graber-Harris-Starr extend to the orbifold context as well. See [Ca 07, §5] for details, and Corollary 9.16 for a very partial positive answer. 9.E. Orbifold uniruledness and canonical dimension. In this subsection we state a fundamental conjecture. It reads as follows. 9.10. Conjecture. One has 1. (X|Δ) is uniruled if κ(X|Δ) = −∞. 2. (X|Δ) is rationally connected if κ+ (X|Δ) = −∞. The converse statements are easily shown to be true. Contrary to the case Δ = 0, it is not known that part 1 of this conjecture implies its part 2. 19 That

is, the largest positive integer s such that KX ∈ {s Pic(X)}.

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The only known case is when n = 2 and Δ logarithmic, by [KMcK 99]. It would be interesting (and probably feasible) to show that their result holds for ﬁnite multiplicities as well. 9.11. Example. Let (P2 |Δ) be the Fano example of 9.3(5). It should be RC, by the preceding conjecture. See 9.3(5) for a possible direct veriﬁcation. It would be interesting to have a deformation-theoretic proof of this property. Notice that such parametrised orbifold rational curves would give, for generic complex numbers u, v, w, a solution to the equation uP 3 + vQ3 + wR5 = S 7 for complex polynomials P, Q, R, S of respective degrees at most 35, 35, 21, 15. 9.F. Global quotients: lifting and images of rational curves. 9.12. Deﬁnition. Let (X|Δ) be smooth. Let g : X D → (X|Δ) be ﬁnite, with X D smooth, and ramifying at least (resp. exactly) above Δ. This means that g is ´etale over the complement of Supp(Δ), and ramiﬁes at order mDj , with mDj a multiple of mj (resp. mDj = mj ) above each Dj . We say that (X|Δ) is a global quotient if there exists g : X D → (X|Δ) with X D smooth, g ﬁnite and ramifying exactly above Δ. %j=3 9.13. Example. Let X = P2 , Δ = j=1 (1 − 1/mj ) Dj , with integral mj > 1, and Dj the lines of equation Tj = 0, in the homogeneous coordinates (T1 , T2 , T3 ). Set m := lcm{m1 , m2 , m3 }. Let f : P2 → P2 be deﬁned by f (Uj ) = Tj := Ujm , for j = 1, 2, 3. Then this morphism ramiﬁes at least (resp. exactly) above Δ for any choice of the mj ’s (resp. if mj = m for all j). 9.14. Remark. If (X|Δ) is a global quotient with Δ K= 0, then it is not simply connected (more precisely, its orbifold fundamental group must have order divisible by the lcm of the mj ’s. For example, P2 with Δ consisting of 4 lines with multiplicities either (3, 3, 5, 7), or (2, 3, 7, 41) is not a global quotient: the orbifold fundamental groups have cardinalities 3 and 1 respectively. 9.15. Proposition. ([Ca 07]) Assume that (X|Δ) is a global quotient, as above. Then one has 1) For each rational curve RD ⊂ X D , R := g(RD ) is Δ-rational. 2) For each Δ-rational R ⊂ X, each component RD of g −1 (R) is rational. 9.16. Corollary. If (X|Δ) is a global quotient, we have (among many other similar statements): 1) (X|Δ) is uniruled (resp. RC) iﬀ so is X D . 2) (X|Δ) is RC iﬀ it is RCC (i.e. if any two generic points of X are connected by a chain of Δ-rational curves). 3. If (X|Δ) is Fano, it is RC. 4. If C ⊂ X is a curve such that −(KX +Δ)·C < 0 through a ∈ (X −Supp(Δ)), there exists a Δ-rational curve through a. 9.17. Example. Let (X|Δ) be as in Example 9.13. By Corollary 9.16, (X|Δ) is RC. Notice that such Δ-rational curves are usually highly singular.

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9.18. Remark. The situation above is very rare. Its interest is however to show that if the deformation theory of rational curves on smooth DM stacks could be developed to the same point as on manifolds, the same results could be obtained for Δ-rational curves on smooth orbifolds, not necessarily global quotients, as when Δ = 0, by replacing the manifold X D above by the smooth DM stack X → X associated to (X|Δ). The following example shows however that some new ideas or techniques seem to be needed. 9.19. Example. Miyaoka-Mori’s Bend and Break does, however, not hold in general for arbitrary smooth, ﬁnite, integral projective orbifolds (X|Δ). Let us indeed choose an isotrivial family of elliptic smooth plane cubics going through a point a ∈ P2 , and degenerating to a union of 3 lines DjD , j = 1, 2, 3 (take for example x3 + y 3 + s = 0 in aﬃne coordinates (x, y), s being a parameter). Let us now blow-up the line D1D , containing the point a, in 3 (or more) of its generic points, none of which is a, obtaining so the manifold X, with three (or more) (−1)-curves Ek , k = 1, 2, 3. We equip now each of the Ek ’s with a large multiplicity m (at least 3 is suﬃcient), obtaining so the orbifold divisor Δ := (1 − 1/m) (E1 + E2 + E3 ) on X. The strict transform D1 in X of the line D1D is thus the only rational curve produced by Miyaoka-Mori’s Bend and Break process through the point a. But this is not a Δ-rational curve, since it intersects transversally each of the Ek ’s. Observe that the orbifold (X|Δ) is not a global quotient, since the complement of the support of Δ is simply connected. 9.G. Δ-rational curves vs ΔQ -rational curves. Let (X|Δ) be a smooth integral projective orbifold. We have two sets of rational curves on X: ﬁrst the set RatlZ (X|Δ) consisting of the ‘divisible’ ones, and the set (see footnote in §9.A for the deﬁnition) RatlQ (X|Δ) of ‘ΔQ -rational curves’ on X (with respect to which one can deﬁne as in 9.2 the notions of Q-uniruledness and Q-rational connectedness). Obviously, RatlZ (X|Δ) ⊂ RatlQ (X|Δ). Examples easily show that the inclusion is strict in general. However, we have the following: 9.20. Conjecture. 1) If (X|Δ) is Q-uniruled (resp. Q-RC), it is uniruled (resp. RC). 2) If (X|Δ) is of general type, there exists a closed proper algebraic subset of X containing all Q-rational curves of (X|Δ). Concerning 9.20.(2), see [PR 06] for the case of general hypersurfaces of the projective space, and [Be 95] for the surface case, by an arithmetic approach. (I thank A. Levin for this reference and an interesting discussion on this topic, leading to the second part of Conjecture 9.20 above).

10. Some Relations with LMMP and the Abundance Conjecture orb The objective is to show how to deduce (when X is projective) Conjectures Cn,m and 5.12 respectively from the two standard conjectures of the LMMP: a weak

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version of the abundance conjecture, and the existence of log-minimal models in the log-canonical case. A third property is actually needed: the stability of the logarithmic cotangent sheaf for log-canonical pairs with ﬁrst Chern class zero. I thank M. P˘ aun for explaining me the notion of numerical dimension. We assume in this section that X is a complex projective connected, ndimensional Q-factorial normal space, and that A and D are Q-divisors on X, with A ample. The following invariant is closely related to the asymptotic Zariski decomposition deﬁned by N. Nakayama ([N 04]) and S. Boucksom ([Bo 02]). The numerical dimension of D is deﬁned as being the following invariant: 0 (X, mD+kA)) ν(X, D) := limk→+∞ {limsupm>0 Log(h Log(m) }, for m > 0 integral and suﬃciently divisible. Easy standard arguments show that: 1) ν(X, D) is either −∞, or lies between 0 and n. 2) ν(X, D) does not depend on the choice of A. 3) ν(X, D) ≥ κ(X, D). 4) ν(X, D) = −∞ if and only if D is not pseudo-eﬀective (this is one of the deﬁnitions of pseudo-eﬀectivity). One (weak) form of the so-called ‘Abundance Conjecture’ is the following: 10.1. Conjecture. Assume (X|D) is a ‘log-canonical pair’. ν(X, KX + Δ) = κ(X, KX + Δ).

Then one has

This is known20 when D is ‘big’ and (X|D) is klt ([BCHM 06], [Pa 08]). This is also known when ν(X, KX + Δ) = 0 if q(X) = 0, it follows from [N 04]and [Bo 02]; when moreover Δ = 0, the general case follows from [CPe 05, §3]. orb is true. 10.2. Proposition. Assume Conjecture 10.1. Then Conjecture Cn,m

Proof: Let f : X → Y be a neat ﬁbration with (X|Δ) a smooth orbifold, with κ(Xy , KXy + Δy ) ≥ 0, for y ∈ Y general. Let (Y |ΔY ) be the orbifold base of (f |Δ) (i.e. Δ(f |Δ) := ΔY ). Then f∗ (m(KX/(Y |ΔY ) + Δ)) is weakly positive for m large and divisible, by Theorem 5.4. We assume that κ(Y, KY + Δ(f |Δ) ) ≥ 0, otherwise the statement is trivial. Thus m(KX + Δ) + f ∗ (B) = m(KX/(Y |ΔY ) + Δ) + f ∗ (KY + ΔY ) + f ∗ (B) is eﬀective for any Q-ample divisor B on Y . And so KX + Δ is pseudo-eﬀective, hence eﬀective, by Conjecture 10.1. It then follows from easy arguments (see for orb example [AC 08, 2.3-4, p. 516]) that Cn,m subadditivity holds in this case. The standard second conjecture of the LMMP is (see [Bi 07, Conjecture 1.1], for example): 10.3. Conjecture. Let (X|Δ) be a log-canonical pair. There exists a sequence of divisorial contractions and ﬂips s : X ''( X D such that, if s∗ (Δ) := ΔD , then (X D |ΔD ) is log-canonical, Q-factorial, and either: 1) KX " + ΔD is nef, or: 2) There exists a ﬁbration f : X D → Y D with −(KX " + ΔD ) relatively ample. 20 Even

for R-divisors.

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This conjecture is known in the klt case, by [BCHM 06], and also when n := dim(X) ≤ 3, by [Sh 06], see also [Bi 07] and the references there. 10.4. Conjecture. Assume (X|Δ) is a ‘log-canonical pair’ with c1 (X|Δ) = 0. Then: 1) κ++ (X|Δ) = 0, deﬁning the left-hand side as being κ++ (X D |ΔD ), for any log-resolution (X D |ΔD ) of (X|Δ). This is implied by the following orbifold version of Miyaoka’s semipositivity theorem: 2) if C is the complete intersection of large multiples of given ample divisors Hi of X, and contained in the open set where (X|Δ) is smooth, then the restriction of S N Ωp (X|Δ) to C should be (H1 , H2 , . . . , Hn−1 )-semi-stable, for any p > 0 and N > 0. This conjecture is known when Δ = 0, using either the existence of Ricci-ﬂat metrics (constructed by S.T. Yau), or Miyaoka’s generic semi-positivity theorem. The same ﬁrst approach seems accessible in the klt case. The log-canonical case might require new ideas. The orbifold version of Miyaoka’s semipositivity theorem might be shown by extending the arguments to the orbifold case, using the orbifold rational curves introduced in §9. orb , 10.4, and 10.3 hold. Then Con10.5. Theorem. Assume that Conjectures Cn,m jecture 5.12 holds. Thus Conjecture 5.12 follows from Conjectures 10.1, 10.4, and 10.3 of the LMMP. When (X|Δ) is smooth, ﬁnite with X projective, Conjecture 5.12 follows from Conjectures 10.1 and 10.4, since Conjecture 10.3 is known from [BCHM 06].

Proof: Let (X|Δ) be a smooth orbifold with X projective. Combining Conjectures 10.4 and 10.3, we see that Conjecture 5.12 holds when κ(X|Δ) = 0. Let F ⊂ ΩpX be a rank-one coherent subsheaf. There are now 2 exclusive cases: either κ(X|Δ) ≥ 0, or κ(X|Δ) = −∞. In the ﬁrst case, let f : X → Y be the Iitaka ﬁbration associated to (KX + Δ). We can assume that f is regular, by making an orbifold elementary modiﬁcation, which does not change the birational invariants we are interested in. Thus dim(Y ) = κ(X|Δ), and κ(Xy |Δy ) = 0 over the generic ﬁbre Xy of f . By a further orbifold modiﬁcation, we can assume that the saturation of F ⊗N in S N Ωp (X|Δ) is locally free for a suitable N such that κ(X|Δ, F) is given by the linear system associated to this saturation. The ﬁltration introduced in the ﬁrst step of the proof of Proposition 4.23 shows, since Conjecture 5.12 holds for (Xy |Δy ), that κ(Xy , F) ≤ 0. The easy addition theorem implies that κ(X|Δ, F) ≤ κ(Xy , F)+dim(Y ) ≤ dim(Y ) = κ(X|Δ), which is the claim in this case. It remains to deal with the case when κ(X|Δ) = −∞. Applying Conjecture 10.3, we get a birational sequence s : (X|Δ) ''( (X D |ΔD ) and a log-Fano ﬁbration f : (X D |ΔD ) → Y , with dim(Y ) < n and (X D |ΔD ) log-canonical. Let H D be a general member of m(−(KX " /Y + ΔD ) + f ∗ (A)), for A suﬃciently ample on Y , and m suﬃciently large, so that (X D |ΔD + (1/m)H D ) := (X D |ΔDD ) is log-canonical, with (KXy" + ΔDDy ) trivial on the generic ﬁbre Xy of f . Choosing m suﬃciently large, and

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making a suitable orbifold modiﬁcation of (X|Δ), we can moreover assume that s : (X|Δ) → (X D |ΔD ) and s : (X|Δ+ ) → (X D |ΔDD ) are regular and log-resolutions, where Δ+ := Δ + (1/m)H, with H being the strict transform of H D in X. Considering the restriction sy : Xy → XyD of s over a generic y ∈ Y , we see that sy : (Xy |Δy ) → (XyD |ΔDy ) is a log-resolution with −(KXy" + ΔDy ) ample on XyD . From Lemma 10.6 below we deduce that κ++ (Xy |Δy ) = −∞ , and Lemma 10.7 then shows that any non-zero section w of F is of the form w = f ∗ (v), for some section v of S N Ωq (Y D |ΔY " ), where (Y D |ΔY " ) is the smooth orbifold base of some suitable neat model of f ◦ s : (X|Δ) → Y , and 0 ≤ q ≤ p. Finally, Lemma 10.8 below implies that q = 0, that is: κ++ (X|Δ) = −∞ if κ+ (X|Δ) = −∞, which is the claimed implication. 10.6. Lemma. Let g : (X|Δ) → (X D |ΔD ) be a birational map from the smooth orbifold (X|Δ) to the log-canonical Fano pair (X D |ΔD ) such that f∗ (Δ) = ΔD . Assume that Conjecture 10.4.(2) holds. Then, for any polarisation of X D , and any general Mehta-Ramanathan curve C ⊂ X D , identiﬁed with its strict transform on X, the following properties hold: 1) For any ﬁnite sequence of pairs of positive integers (Nh , qh ), h = 1, . . . , t, and any subsheaf F ⊂ S N1 Ωq1 (X|Δ) ⊗ · · · ⊗ S Nt Ωqt (X|Δ), the restriction F|C has negative degree. In particular, H 0 (X, S N1 Ωq1 (X|Δ) ⊗ · · · ⊗ S Nt Ωqt (X|Δ)) = {0}. 2) κ++ (X|Δ) = −∞. Proof: Let G := S N1 Ωq1 (X|Δ) ⊗ · · · ⊗ S Nt Ωqt (X|Δ), and let L be a line bundle of degree 0 on C. It is suﬃcient (by considering Λr G, for any r > 0) to show that the degree of any rank one coherent of G is negative over C, and even that %subsheaf j=n H 0 (C, G ⊗ L) = 0. Let H D = j=1 Hj , where the Hj ’s are general members of m(−(KX " + ΔD )), m > 0 being a suﬃciently large integer, the Hj ’s being chosen so that (X D |ΔD + (1/mn)H D ) := (X D |ΔDD ) is log-canonical, with (KX " + ΔDD ) trivial on X D , and such that Δ+ := (Δ + (1/mn)H) has normal crossings support21 , H being the strict transform of H D in X. Choose C ⊂ X D to be a generic curve cut out by Mehta-Ramanathan very ample linear systems, and meeting H D transversally, but not meeting the indeterminacy locus of g −1 , and so identiﬁed with its strict + transform on X, then any non-zero section of L ⊗ GC := L ⊗ S N1 Ωq1 (X|Δ+ ) ⊗ Nt qt + · · · ⊗ S Ω (X|Δ )|C has no zero at all on C, since it generates a rank one + subsheaf of the locally free sheaf L ⊗ GC , assumed to be semi-stable, and with trivial determinant. Assume now that w is a non-zero section %j=tof L ⊗ G. Replacing w by a suitable tensor power, we may assume that N := j=1 Nh qh ≥ n/p(1 − 1/mn). Because of + , and has thus no zero the natural inclusion G ⊂ G + , w is also a section of L ⊗ G|C + (as a section of L ⊗ G|C ). Now choose a generic point a ∈ X D such that w(a) K= 0 as a section of L ⊗ G|C . We choose a outside of the support of ΔD and belonging to the smooth locus of X D . On the other hand, since the Hj can be chosen arbitrary generically, we may assume that they build a system of coordinate hyperplanes 21 Thus

µ=

Δ+ is not integral (unless m = n = 1), even if Δ is. So Q-multiplicities (here equal to 1 = 1 + mn−1 ) are needed in Conjecture 10.4 in the present argument.

mn mn−1

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for suitable local coordinates at a. Now we check immediately, using Lemma 3.3 and multilinearity, that in the local OX " -basis dz(J) of G + described in §3.B, all coordinates of w vanish at a. This contradicts the non-vanishing property above, and proves the claims. 10.7. Lemma. Let (X|Δ) be a smooth Fano orbifold, and f : X → Y be a ﬁbration with generic smooth orbifold ﬁbre (Xy |Δy ). Assume that, for any ﬁnite sequence of pairs of positive integers (Nh , qh ), h = 1, . . . , t, one has H 0 (Xy , S N1 Ωq1 (Xy |Δy ) ⊗ · · · ⊗ S Nt Ωqt (Xy |Δy )) = {0}. Then any section w of S N Ωp (X|Δ) is of the form w = f ∗ (v), for some section v of S N Ωq (Y D |ΔY " ), where (Y D |ΔY " ) is the smooth orbifold base of some suitable neat model of f ◦ s : (X|Δ) → Y , and for some q with 0 ≤ q ≤ p. In particular: κ++ (X|Δ) ≤ dim(Y ). Proof: This is just Lemma 4.23. 10.8. Lemma. Let (X|Δ) be a smooth orbifold with κ+ (X|Δ) = −∞, and let f : X → Y be a ﬁbration. If (Y D |ΔY " ) is the orbifold base of some suitable neat model of f such that (Y D |ΔY " ) is smooth, then κ+ (Y D |ΔY " ) = −∞. Proof: After an orbifold modiﬁcation of (X|Δ), we can assume that f is an orbifold morphism, in which case f ∗ (S N Ωp (Y D |ΔY " )) → S N Ωp (X|Δ) is well-deﬁned for any N, p ≥ 0, so that κ+ (Y D |ΔY " ) ≤ κ+ (X|Δ) = −∞. orb and 5.12 10.9. Corollary. The conjectures 10.1, 10.3 and 10.4, and thus Cn,m are known (see, for example, [K 92] and [Bi 07]) in the following special cases: 1) n ≤ 2, 2) n = 3 and Δ = 0.

11. Some Conjectures 11.A. Lifting of properties. We assume in this §11.A the existence of the ‘κorb rational quotient’ for smooth projective orbifolds (this is true if so is Cn,m ). Thus n we have the decomposition c = (M ◦ r) for any such orbifold. Then we have the following. 11.1. Proposition. Let P be a property of smooth projective orbifolds which is: 1) birational. 2) satisﬁed when κ = 0 and κ+ = −∞. 3) stable by exensions (i.e. satisﬁed by (X|Δ) if satisﬁed by the general orbifold ﬁbre and the orbifold base of some neat ﬁbration (f |Δ) : (X|Δ) → Y ). Then P is satisﬁed by any special orbifold. Assume moreover that the property P : 4) is not satiﬁed by the positive-dimensional orbifolds of general type. 5) is satisﬁed by (X|Δ) if satisﬁed by (X D |ΔD ), and there exists a surjective orbifold morphism (X D |ΔD ) → (X|Δ). Then P is satisﬁed exactly by the special orbifolds.

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11.B. The case of special orbifolds. Qualitatively special orbifolds should behave like rational and elliptic curves and they are deﬁned similarly: coherent rank 1 subsheaves L ⊂ ΩpX should have κ(X|Δ, L) < p, for any p > 0. We formulate the conjectures in general, although the invariants below have not been deﬁned when Δ K= 0 here. 11.2. Conjecture. Let (X|Δ) be a smooth projective orbifold. • If (X|Δ) is special, then π1 (X|Δ) is almost abelian. • (X|Δ) is special if and only if its Kobayashi pseudometric d(X|Δ) ≡ 0. • (X|Δ) (deﬁned over a number ﬁeld) is special if and only if potentially dense. • If (X|Δ) is special, so are its deformations (see [Ca 07] for a precise statement). 11.3. Remark. 1) The motivation for these conjectures is of course 11.1, which might give a possible approach for a proof: 2) The conjecture should hold for the crucial cases: κ(X|Δ) = 0 and κ+ (X|Δ) = −∞, which might be proved ﬁrst by other speciﬁc methods for the particular cases KX + Δ < 0 and c1 (KX + Δ) = 0 (in the case of π1 , for example, one may think of L2 theory, and Ricci-ﬂat metrics). See also Remark 11.8.(1). 3) The properties should be shown to be preserved by ‘extensions’ (i.e. should hold for (X|Δ) if they hold for the orbifold ﬁbres and base of a ﬁbration f : X → Y ). Notice that the consideration of the orbifold structure on the base precisely means that local obstructions to lifting vanish (in codimension one at least). The expected ‘extension’ property is thus a kind of ‘local-to-global’ principle. This extension property holds for the orbifold fundamental group (see orb , the conclusion would follow from Theorem 8.6. [Ca 07], §12). 4) Assuming Cn,m 5) One might even wonder, in case of the second conjecture above, whether being special is not equivalent to have any two points joined by an entire (transcendental) orbifold curve h : C → X. 6) The hyperbolicity and arithmetic conjectures above have an obvious function ﬁeld analogue. See [Ca 07] for details. 11.C. The general case. The general case should be ‘split’ into its two antithetical parts (special and ‘general type’) by the core. See [Ca 07] for deﬁnitions and details. 11.4. Conjecture. Let (X|Δ) be a smooth projective orbifold, and let c : (X|Δ) → C(X|Δ) = C be its core. Then: • d(X|Δ) = (c)∗ (δ), with δ := d(C|Δ(c|Δ)) , the Kobayashi pseudometric on the base orbifold of (c|Δ), which is a metric on some Zariski dense open subset U of C. • If (X|Δ) is deﬁned over a number ﬁeld, then (U ∩ c((X|Δ)(k))) is ﬁnite, for any number ﬁeld of deﬁnition of (X|Δ), and any ‘model’ of (X|Δ). • The ﬁbration c deforms under deformations of (X|Δ), in particular dim C(X|Δ) is constant in such a deformation. 11.5. Remark. Let us motivate the ﬁrst conjecture above (the second one is similar): since the general orbifold ﬁbres of c are special, their Kobayashi pseudometric vanishes, after 11.2. Thus d(X|Δ) = (cX )∗ (δ) for some δ on C. By the deﬁnition of

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Kobayashi pseudometric on the orbifold base of c, we have δ ≥ d(C|Δ(c|Δ)) . The reversed inequality should then follow from the fact that, locally on the complement of a 2-codimensional subset of C(X), the orbifold maps h : D → (C(X)|Δ(X|Δ)) lift to X. The assertion that δ is a metric generically on C(X|Δ) is simply an orbifold version of Lang’s conjecture. Similarly for the ﬁniteness in the arithmetic conjecture. 11.D. Families of canonically polarised manifolds. When (X|Δ) is a smooth logarithmic orbifold, the core and the notion of specialness produce new invariants of the quasi-projective manifold U = X − Δ, independent from its smooth projective compactiﬁcations. This framework appears to be suitable also in moduli problems. 11.6. Conjecture. (“Isotriviality Conjecture”) We conjecture (in [Ca 07], §12.6) that algebraic families of canonically polarised manifolds parametrised by special quasi-projective varieties X 0 are isotrivial, and so that, for any quasiprojective base, the moduli map factors through the ‘core’ of the base of this base. This extends and uniﬁes previous conjectures by Viehweg-Zuo ([VZ 02]) and Kebekus-Kov´ acs (see [JK 09] and [Ca 07, §12.6] for details). This conjecture is proved in dimensions 3 or less in [JK 09] (see also relevant references there). In fact, this conjecture can be reduced to other conjectures of classiﬁcation, using the Viehweg-Zuo sheaves in [VZ 02]: 11.7. Proposition. The conjecture above (i.e. algebraic families of canonically polarised manifolds parametrised by special quasi-projective varieties are isotrivial) orb , 10.3 and 10.4 hold for dimensions is true when dim(X 0 ) = d if Conjectures Cn,m at most (d − 1). This conjecture is thus, by the observations made in §10, a consequence in dimension d of Conjectures 10.1, 10.3 and 10.4 in dimension at most (d − 1). Proof: It essentially consists in checking the properties 1-3 listed in Proposition 11.1. Let X 0 = (X − D) be a smooth logarithmic compactiﬁcation of the base space X 0 of this family. Then c = (M ◦ r)n is the constant map to a point, since X 0 is special and n-dimensional. It is thus suﬃcient to show, inductively on n, that if µ : X 0 → M is the moduli map to the coarse moduli stack induced by the given family (and constructed in [Vieh 95], Theorem 1.11), that µ factorises through f if (f |D) : (X|D) → Y is a ﬁbration with general orbifold (logarithmic) ﬁbres having either κ = 0, or κ+ = −∞. In other words: it is suﬃcient to show the conjecture when κ(X|D) = 0, and when κ+ (X|D) = −∞. We now assume one of these properties to hold, and also that Var(g 0 ) > 0, where Var(g 0 ) is the generic rank of the map µ, or equivalently, of the KodairaSpencer map associated to the family of canonically polarized manifolds under consideration. The main theorem of [VZ 02] asserts the existence of a line bundle A ⊂ SymN (Ω1X (log D)) with κ((X|D), A) ≥ Var(g 0 ). An important reﬁnement of this result ([JK 09], Theorem 9.3), shows that A ⊂ SymN (B) for some subsheaf

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B ⊂ Ω1X (log D), generically of the form B = S(dµ : µ∗ (Ω1M ) → Ω1X 0 ). Using now the arguments of the proof of Theorem 10.5, we get: κ(X|D, A) ≤ κ++ (X|D) = κ(X|D) ≤ 0, contradicting our hypothesis that Var(g 0 ) > 0. 11.8. Remark. 1. Conjecture 5.12 is needed only for the cases where κ+ = −∞ or κ = 0. The ﬁrst case is a consequence of Conjecture 9.10, and the second might be solved using LMMP to reduce to the case of a trivial ﬁrst Chern class, possibly accessible by constructing orbifold Ricci-ﬂat K¨ ahler metrics. See §10 for details. 2. The meaning of the conjecture above is also that any subvariety of the moduli stack M should be of logarithmic general type. In particular, according to Conjecture 7.1, the logarithmic Kobayashi pseudometrics of these subvarieties should be generically metrics. A statement in this direction is shown in [VZ 03]. If one could prove that any two generic points of a special orbifold can be connected by chains of orbifold entire curves h : C → (X|Δ) (see Remark 11.3.(5)), the conjecture would also follow from [VZ 03]. 3. Since Conjectures 10.1,10.3 and 10.4 are known in dimension at most 2, and also in dimension 3 if Δ = 0, we see that the isotriviality conjecture is true when d = 3 (as shown in [JK 09]), and also when d = 4 if Δ = 0.

12. Special versus Weakly Special Manifolds In this section, we illustrate by an example the diﬀerence between the two notions, and its implications in arithmetics and hyperbolicity questions. It also shows that orbifold structures need to be considered in birational classiﬁcation. Recall (Deﬁnition 6.9) that the (complex, projective, connected) manifold X is ‘weakly special’ if none of its ﬁnite ´etale covers maps rationally onto a positivedimensional variety of general type. Special X are weakly special, and conversely for curves and surfaces. However, simply connected threefolds which are weakly special, but not special, are constructed in [BT 04] (see Remark 6.11 above). Some of their examples are deﬁned over Q. We have the following conjecture, due to Abramovich and ColliotTh´el`ene, stated in [HT 00]: 12.1. Conjecture. (Abramovich-Colliot-Th´el`ene) Let X be deﬁned over a number ﬁeld. Then X is potentially dense if and only if X is weakly special (as formulated in our terminology). The ‘only if’ part follows from Lang’s conjecture and Chevalley-Weil’s theorem. Observe that this conjecture conﬂicts with item 3 of Conjecture 11.2 for Bogomolov-Tschinkel threefolds X (if deﬁned over Q, say). Indeed, 12.1 above claims that X is potentially dense, while our conjecture 11.4 claims they are not (and even that cX (X(k)) ∩ U is ﬁnite for any number ﬁeld k). Let us sketch the construction of the Bogomolov-Tschinkel threefolds given in [BT 04].

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12.2. Example. One needs the following ingredients: 1. A smooth elliptic surface g : T → P1 with exactly one multiple ﬁbre m.T0 = ∗ g (0) of multiplicity m ≥ 2, with T0 smooth connected, such that (T − T0 ) is simply connected. Such a surface can be constructed by a logarithmic transform on a suitable elliptic rational surface. 2. A smooth simply connected elliptic surface hD : S D → P1 with κ(S D ) = 1, together with an ample base-point free line bundle LD , and a generic pencil of (generically smooth) sections of LD giving a map ϕ : S D ''( P1 . By Lefschetz’s theorem, the complement in S D of a smooth member of this pencil has a ﬁnite fundamental group, equal to the order of divisibility of LD in Pic(S D ). Two generic members B, B D of the pencil meet transversally at a ﬁnite set P D of points pDj of S. Blow these points up to get a regular map ψ : S → P1 on the blown-up surface S. Denote by D, DD the strict transforms of B, B D in S. The crucial observation is that their complements in S are simply connected (this is because the lift to S of a small loop around B in S D , and close to pj say, becomes homotopically trivial in S, since already homotopically trivial in (Ej −qj ), if Ej is the exceptional divisor of S above pj , and qj is the intersection point of D with Ej ). The sought-for elliptic threefold f : X → S is then just obtained from ψ : S → P1 by the base change g : T → P1 . It has a multiple ﬁbre of multiplicity m exactly above the ﬁbre D of ψ, so that Δ(f ) = (1 − 1/m)D. From this, one immediately deduces that κ(S) = 1 < κ(S, KS + Δf ) = 2. It only remains only to check that X is simply connected, which easily follows from the simple connectedness of (T −T0 ) and (S − D). The known methods of arithmetic geometry are presently unable to solve the problem of whether these threefolds are potentially dense or not (one has to decide whether certain simply connected smooth orbifold surfaces of general type are potentially dense or not). Instead of this, one can answer the hyperbolic analogue for some examples at least. Let us state the hyperbolic analogue of the Abramovich-Colliot-Th´el`ene conjecture 12.1: 12.3. Question. Is a (complex, projective, connected) manifold X ‘weakly special’ if and only if dX ≡ 0 ? The answer to this question is ‘no’ for the ‘only if’ part (the ‘if’ part follows from Lang’s conjecture). This shows that either Conjecture 12.1 is wrong, or that the expected links between arithmetics and hyperbolicity fail to hold. More precisely: 12.4. Theorem. ([CPa 07]) There exist certain simply connected elliptic threefolds cX : X → S constructed in [BT 04] (and so: weakly special, but not special) such that, for any entire curve h : C → X, the composed map cX ◦ h : C → S is either constant, or has image contained in some ﬁxed projective curve C ⊂ S (independent of h). ([CW 05]) Certain of the examples above can be chosen so that the curve C is empty. In this case, dX = (cX )∗ (δ), for some continuous metric δ on S (which establishes Conjecture 11.4 in these cases).

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The proof consists in adapting to the orbifold context the method of Bogomolov to show that surfaces of general type with (c21 − c2 ) > 0 contain only ﬁnitely many special curves, as extended to the transcendental case by McQuillan. The methods are applied to the (general type) orbifold base (S|ΔcX ) of cX to show that all orbifold maps hD : C → (S|ΔcX ) are algebraically degenerate. Because f ◦ h is such an orbifold map for any h : C → X, the conclusion follows. In [Rou 06] this result is generalised, and its proof simpliﬁed, using some further results of McQuillan.

13. Classical versus ‘Non-Classical’ Multiplicities The introduction of multiplicities deﬁned by inﬁmum rather than gcd creates great technical diﬃculties, although possibly not modifying the qualitative geometry. We illustrate this by two examples from [Ca 05]. See also Conjecture 9.20. We ﬁrst start (see 13.1 below) with an example showing a huge discrepancy in the case of ﬁbres of general type case. We ask then (see Questions 13.2 and 13.3) whether this discrepancy can occur when the ﬁbres are special. 13.1. Theorem. There exist general type ﬁbrations f : S → P1 with S a smooth projective connected and simply connected surface. Note that the orbifold base cannot be the ‘classical’ (or ‘divisible’) orbifold base. Indeed, if f : S → P1 were of general type for the ‘classical’ multiplicities, a ﬁnite ´etale cover of S would map onto a hyperbolic curve, and its fundamental group would have the free group on two generators as a quotient. Note also that the smooth ﬁbres of f are hyperbolic curves, since multiple ﬁbres of elliptic surfaces are always ‘divisible’ (i.e. inf and gcd coincide for them). In fact in the examples of [Ca 05], the genus of the ﬁbres is 13. Other more interesting examples with ﬁbres of any genus at least 2 have been constructed by Lidia Stoppino in [Sto 06]. Finally, observe that S is necessarily of general type (by orbifold additivity, for example). Let us give a brief sketch of the construction of [Ca 05]: for suitable choice of ﬁve distinct lines Tk , k = 1, 2 and Dj , j = 1, 2, 3 of P2 meeting in a point a, we assign to the lines Dj the multiplicity 2, and to the lines Tk the multiplicity 3 getting a (non-reduced) curve C of degree 2 · 3 + 3 · 2 = 12, we show the existence of an irreducible curve C D ⊂ P2 of degree 12 not going through a, and meeting each of the Tk (resp. Dj ) in 4 (resp. 6) distinct points with order of contact exactly 3 (resp. 2). The pencil of curves of degree 12 on P2 generated by C, C D gives a rational map hD : P2 → P1 which we resolve as h : P → P1 . One checks then that it has a (non-classical) multiple ﬁbre C0 (of multiplicity 2) consisting of a (simply connected) tree of rational curves with multiplicities either 2 or 3. To get f : S → P1 , one just makes a base change through a map r : P1 → P1 of degree at least 5 ramiﬁed at generic points. The simple connectedness of S follows from the exact sequence of groups: π1 (F ) → π1 (S) → π1 (P1 ), if F is a generic ﬁbre of f . The exactness of this sequence follows because f has no classical multiple ﬁbre;

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the image of π1 (F ) in π1 (S) is trivial because f has some simply connected ﬁbres (like C0 ). This ﬁnishes the sketch of the construction. Some of the examples above are deﬁned over Q. For any number ﬁeld k, the quantity f (S(k)) should be ﬁnite after Conjecture 11.4, which would establish Lang’s conjectural ‘mordellicity’22 for them, and give the ﬁrst example of a ‘mordellic’ smooth simply connected surface of general type. Because f (S(k)) is contained in the set of rational points of the base orbifold (P1 |Δf ) of f , which is of general type, the problem would be solved if one could establish the orbifold Mordell conjecture 14.2 below. 13.2. Question. Let f : X → B be a ﬁbration with X smooth, and generic (smooth) ﬁbres F having κ(F ) = 0. Do we have the equality: Δf = Δ∗f ? (In other words: do the inf and gcd-multiplicities then coincide ?) This holds when F are abelian varieties. The ﬁrst non-trivial case is for K3 surfaces (and even Kummer surfaces). More generally: 13.3. Question. 1) Let f : X → B be a ﬁbration with X smooth, and generic (smooth) ﬁbres special. Do we have the equality: Δf = Δ∗f ? (Observe indeed that the property holds when F is rationally connected, by [GHS 03]). 2) If X is a ‘classical’ special manifold, is it special? (Being ‘classically’ special means that there is no neat (rational) ﬁbration on X with classical orbifold base of general type.)

14. An Orbifold Version of Mordell’s Conjecture We shall introduce two notions (‘classical’ and ‘non-classical’) of Q-rational points on certain orbifolds of the form (P1 |Δ). These two notions are deduced from the two natural notions (based either on gcd or inf) of orbifold morphisms, when rational points are seen as sections of the arithmetic surface P1 over the spectrum of the ring of integers. These two notions are also respectively compatible functorially with the two natural notions of ﬁbre multiplicities considered in the text in the following sense: if f : S → P1 is a ﬁbration with orbifold base (resp. ‘classical’ orbifold base) (P1 |Δ), everything deﬁned over Q, then f (X(Q)) is contained in the set of Q-rational (resp. ‘classical’ Q-rational) points of (P1 |Δ) deﬁned below. Let (P1 |Δ) be a geometric orbifold, with integers p, q, r > 1, and Δ := (1 − 1/p) {0} + (1 − 1/q) {1} + (1 − 1/r){∞} Then one has κ(P1 |Δ) = 1 if and only if (1/p + 1/q + 1/r) < 1. This is the case for example if (p, q, r) = (2, 3, 7). This geometric orbifold is deﬁned over k = Q. 22 i.e.

S(k) ∩ U is ﬁnite for any k and some ﬁxed Zariski open nonempty U ⊂ S.

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Let us now deﬁne its Q-rational points, and also their ‘classical’ version. The set (P1 |Δ)(Q)∗ of ‘classical’ Q-rational points of (X|Δ) consists of the usual rational points x = αp /β r , with α, β ∈ Z, gcd(α, β) = 1 such that there exists γ ∈ Z and αp + β r = γ q . These points are the sections of the arithmetic surface P1 over the spectrum of the ring of integers, meeting the three sections deﬁned by the points 0, ∞, 1 with arithmetic orders of contact divisible by p, q and r respectively. Using Faltings’s and Chevalley-Weil’s theorems, one gets: 14.1. Theorem. ([DG 95]) (‘Faltings+ε’) (P1 |Δ)(Q)∗ is ﬁnite if κ(P1 |Δ) = 1. The set (P1 |Δ)(Q) of (‘non-classical’) Q-rational points of (P1 |Δ) consists of the usual rational points x = a/b with a, b ∈ Z, (a, b) = 1 such that a is ‘p-full’, b is ‘r-full’, and c := a − b is ‘q-full’, where a is ‘p-full’ means: for any prime ] dividing a, then: ]p divides a. Notice that, by a result of Erd¨ os, the set of p-full integers behaves asymptotically as the set of p-th powers: Card{a ≤ X|a is p-full} ∼X→+∞ Cp X 1/p for a certain explicit constant Cp > 0. These points are the sections of the arithmetic surface P1 over the spectrum of the ring of integers, meeting the three sections deﬁned by the points 0, ∞, 1 with arithmetic orders of contact at least equal to p, q and r respectively. 14.2. Conjecture. (Orbifold Mordell conjecture) (P1 |Δ)(Q) is ﬁnite if κ(P1 |Δ) = 1. 14.3. Remark. The conjecture above is open. It is easy to show23 that the abcconjecture implies the Orbifold Mordell conjecture. This conjecture can be stated similarly (but less concretely) for arbitrary number ﬁelds k. It is a (very) special case of Conjecture 11.4 above.

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Fr´ed´eric Campana E-mail: [email protected]

Birational geometry of threefolds Jungkai Alfred Chen∗

1. Introduction One of the major goals of birational geometry is to ﬁnd a good model inside a birational equivalency class and to study the geometry of such a model. Therefore, it consists of two major parts: the minimal model program and the geometry of minimal models and Mori ﬁber spaces. In the case of surfaces, a satisfactory theory exists already. Given a projective surface X, by contracting (−1) curves, one obtains a model Xmin , which is either a Mori ﬁber space or of Kodaira dimension κ ≥ 0. Surfaces with κ = 0 can be classiﬁed into 4 types according to their irregularity q(X) and genus pg (X). By considering its pluricanonical map, a surface with κ = 1 is an elliptic surface. Thanks to the work of Kodaira on elliptic surfaces, we have a good understanding of surfaces with κ = 1. Kodaira and Bombieri studied surfaces of general type systematically. In fact, Bombieri proved that the m-canonical map ϕm is birational for m ≥ 5 (cf. [2]). Turning to dimension ≥ 3 in general, the situation becomes more involved and is in many aspects more complicated. First of all, extremal contractions usually produce singularities. In fact, one needs to allow terminal singularities in order for the minimal model program to work. In particular, a minimal model might have terminal singularities. This increases the diﬃculty of the study of the geometry of minimal models. Another complicating phenomenon is that one needs a birational surgery called a ﬂip in order to avoid annoying small contractions. The purpose of this article is to give a brief survey of recent work [4, 5, 6, 3, 8] on the birational geometry of threefolds. The main ingredient is the classiﬁcation of terminal singularities in dimension 3, due to the work of Reid and Mori (cf. [28, 29]). Moreover, there is a singular Riemann–Roch formula that counts the contribution of singularities due to Reid (cf. [29]). This article is organized as follows. In Section 2, we recall some facts about terminal singularities in dimension 3. We survey the notions of packing and canonical sequence of prime unpackings in Section 3. The packing of baskets occurs in many geometric maps, such as divisorial contractions and ﬂips. We will give examples instead of proofs. We refer the reader to [4] for the general machinery, where Meng Chen and the author developed some formal properties of baskets. In Sections 4,5 and 6, we review applications to various ﬁniteness and eﬀectiveness problems in Fano threefolds, varieties of general type and weighted complete intersections. ∗ The

author was partially supported by TIMS, NCTS/TPE and the National Science Council of Taiwan.

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Sections 7 to 9 can be regarded as part II, where we try to answer what happens to singularities in the minimal model program. The details of this part can be found in [8]. For this purpose, we introduce the notion of depth of singularities in section 8. We explain the idea of factorizing birational maps by using induction on the depth. We always work on projective normal Q-factorial varieties over the complex number ﬁeld C, unless otherwise stated.

2. Three dimensional terminal singularities Let X be a projective threefold. It is said to be a terminal (resp. canonical) threefold if there is a resolution µ : Y → X such that KY = µ∗ KX +

M

a(X, Ei )Ei ,

with a(X, Ei ) > 0 (resp. a(X, Ei ) ≥ 0) for all µ-exceptional divisors Ei . The index of a singularity (P ∈ X) is deﬁned to be the Cartier index, i.e. the minimal positive integer r such that rKX is Cartier at P . It is well-known that a three dimensional terminal singularity is isolated. A terminal singularity of index 1 is a cDV singularity. That is, P ∈ X is a hypersurface singularity given by an equation of the form ϕ = f (x, y, z) + tg(x, y, z, t) where f (x, y, z) is the equation of a DuVal singularity. Given a terminal singularity (P ∈ X) of index r > 1, there is a canonical cover π : Y → X such that π −1 (P ) ∈ Y is terminal of index 1. In [28], Mori classiﬁed three dimensional terminal singularities of index r > 1. They are classiﬁed into the types cA/r, cAx/2, cAx/4, cD/2, cD/3 and cE/2. A summary of this classiﬁcation can be found in [11], and in [29] as well. Notice that in each case, there exists a coordinate, say t, sharing the same eigenvalue with ϕ. Then the deformation given by ϕλ := ϕ + λt along the direction of t admits a µr -action. One sees that the terminal singularity P ∈ X is deformed into a collection of terminal cyclic quotient singularities Q1 , ..., Qk in Xλ := (ϕλ = 0)/µr . The collection of terminal cyclic quotient singularities Q1 , ..., Qk is called the basket of (P ∈ X). By a 3-dimensional quotient singularity Q of type 1r (a, b, c), we mean a singularity which is analytically isomorphic to the quotient of (C3 , o) by a cyclic group action: ε(x, y, z) = (εa x, εb y, εc z), where r is a positive integer and ε is a ﬁxed r-th primitive root of 1. The Terminal Lemma asserts that Q is terminal if and only if it can be normalized to a singularity of type 1r (1, −1, b) with an integer b coprime to r and 0 < b < r, by replacing ε with another primitive root of 1 and changing the ordering of coordinates. By replacing ε with another primitive root of 1, we may and will assume that b ≤ 2r .

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2.1. Baskets. A basket B of singularities is a collection (allowing multiplicities) of terminal quotient singularities of type r1i (1, −1, bi ), i ∈ I where I is a ﬁnite index set. For simplicity, we will always denote a terminal quotient singularity 1 r (1, −1, b) by a pair of integers (b, r). So we will write a basket as: B := {ni × (bi , ri )|i ∈ I, ni ∈ Z+ }, where the ni denote the multiplicities. Given baskets B1 = {ni × (bi , ri )} and B2 = {mi × (bi , ri )}, we deﬁne B1 ∪ B2 := {(ni + mi ) × (bi , ri )}. Notice that given a terminal singularity (P ∈ X), one can attach a basket B(P ∈ X) to it by considering the types of quotient singularities Q1 , ..., Qk in its deformation Xλ . Notice also that B(P ∈ X) is independent of 0 < |λ| > 1. 2.2. Singular Riemann-Roch formula. Let X be a terminal threefold. In [29], Reid shows that the contribution of a terminal singularity to the RiemannRoch formula can be computed by its basket. Indeed, the precise formula for a pluricanonical divisor mKX can be written as χ(mKX ) =

m−1 M m(m − 1)(2m − 1) 3 KX + (1 − 2m)χ(OX ) + 12 j=1

The correction term is l(j, Q) = rQ , i.e. x = x − rQ \ rxQ L.

jbQ (rQ −jbQ ) , 2rQ

M

l(j, Q).

Q∈B(X)

where x denotes the residue modulo

3. Packings and canonical sequence of baskets In this section, we introduce the notion of packing between baskets of singularities. This notion deﬁnes a partial ordering on the set of baskets. For a given basket, we deﬁne its canonical sequence of prime unpackings. The canonical sequence plays a fundamental role in our arguments. The details can be found in [4, Section 2,3]. Deﬁnition 3.1. Given baskets B and B D . We say that B D is a packing of B (or B is an unpacking of B D ), denoted by B e B D , if one of the following holds. (1) B = {(b1 , r1 ), (b2 , r2 )} ∪ B0 and B D = {(b1 + b2 , r1 + r2 )} ∪ B0 for some basket B0 . (2) B = {(b1 , r1 )} ∪ B0 and B D = {(b1 , r1 + 1)} ∪ B0 for some basket B0 . Deﬁnition 3.2. The packing B = {(b1 , r1 ), (b2 , r2 )} ∪ B0 e B D = {(b1 + b2 , r1 + r2 )} ∪ B0 is called a prime packing if b1 r2 − b2 r1 = ±1. A prime packing is said to have level n if r1 + r2 = n.

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Remark 3.3. Only the packing of the type B = {(b1 , r1 ), (b2 , r2 )} ∪ B0 and B D = {(b1 + b2 , r1 + r2 )} ∪ B0 is the packing deﬁned originally in [4]. We generalize the notion a little bit here. It is easy to verify that this generalization satisﬁes the properties of packings in [4]. The seemingly mysterious notion of packings can indeed be realized in various elementary birational maps. Example 3.4. We consider the Kawamata blow-up [20]. Let X = XΣ be a toric threefold associated to a fan Σ. Suppose that there is a cone σ in Σ generated by v1 = (1, 0, 0), v2 = (0, 1, 0) and v3 = (s, r − s, r) with 0 < s < r and (s, r) = 1. The cone σ gives rise to a quotient singularity (P ∈ X) of type 1r (r − s, s, 1). ˜ → X be the partial resolution obtained by the subdivision by adding Let π : X ˜ has two quotient singularities of type 1 (r, −r, 1), v4 = (1, 1, 1). One sees that X s 1 and r−s (r, −r, 1) respectively, where ¯. denotes the residue modulo s and r − s respectively. ˜ = {(bD , s), (b−bD , r−s)} Then it is easy to verify that B(X) = {(b, r)} and B(X) D D for some b, b satisfying b r − bs = ±1. One sees that ˜ e B(X) B(X) is a prime packing of baskets. Example 3.5. Let X = XΣ be a toric threefold associated to a fan Σ. Suppose that there are two cones σ4 , σ3 in the fan Σ such that σ4 is generated by v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1); σ3 is generated by v1 = (1, 0, 0), v2 = (0, 1, 0), v4 = (s, r − s, −r). with 0 < s < r and (s, r) = 1. Let X + be the threefold obtained by replacing σ4 , σ3 with σ1 , σ2 , where σ1 is generated by v2 , v3 , v4 ; σ2 is generated by v1 , v3 , v4 . The birational map X ''( X + is a toric ﬂip. One can verify that B(X) = {(b, r)} and B(X + ) = {(bD , s), (b − bD , r − s)} for some b, bD satisfying bD r − bs = ±1. Similarly, B(X + ) e B(X) is again a prime packing of baskets. Theorem 3.6. Given a basket B, there is a uniquely determined sequence of unpackings of baskets B (0) (B) # B (5) (B) # ... # B (n) (B) # ... # B satisfying the following properties: (1) B (0) (B) consists of baskets of the type {(1, r)|r ≥ 2}.

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(2) B (n) (B) consists of baskets of the type {(b, r)|b = 1, r ≥ 2, or r ≤ n}. (3) B (0) (B) # B (5) (B) consists of several prime packings of level 5. (4) B (n−1) (B) # B (n) (B) consists of several prime packings of level n for n ≥ 6. (5) B (n) (B) = B for all n ≥ max{ri |(bi , ri ) ∈ B}. Example 3.7. Take a basket B = {(q, p)} with (p, q) = 1 for example. Let [0, a1 , ..., an ] be the continued fraction expression of pq . Notice that [0, a1 , ..., an ] = [0, a1 , ..., an − 1, 1] if an > 1. We may and do assume that an = 1 in the expression of pq . One can deﬁne positive integers hj , kj recursively so that hj = aj hj−1 + hj−2 , kj = aj kj−1 + kj−2 , hj kj = [0, a1 , ..., aj ]. In particular, we have pq = hknn , hn = hn−1 + hn−2 and kn = kn−1 + kn−2 . Hence we have an unpacking {(hn , kn )} ≺ {(hn−1 , kn−1 ), (hn−2 , kn−2 )}. Since hn−1 kn−2 − hn−2 kn−1 = (−1)n , the unpacking is a prime unpacking. We consider B (j) (B) := {(hn−1 , kn−1 ), (hn−2 , kn−2 )} for kn−1 ≤ j < kn = q. Then the sequence B (kn−1 ) (B) = ... = B (q−1) (B) e B (q) (B) = B satisﬁes the above properties. The remaining parts of the sequence can be constructed inductively. The sequence is called the canonical sequence of B. Let rn (B) be the number of prime packings of level n in the sequence, which equals the number of prime packings B (n−1) (B) e B (n) (B). By deﬁnition, one has rn (B) ≥ 0 for all n. This gives rise to many useful inequalities. A typical inequality which we would like to mention here is the following one coming from the fact r10 + r12 ≥ 0 (see [4, Equation 5.3]). 2χ5 + 3χ6 + χ8 + χ10 + χ12 ≥ χ + 10χ2 + 4χ3 + χ7 + χ11 + χ13 ,

(3.1)

where χ := χ(X, OX ) and χm := χ(X, O(mKX )). The above inequality plays a crucial role in our argument, though we have obtained many other similar inequalities. For details of the proof and the combinatorics involved, please see [4, Sections 3,4]. By observing the Riemann–Roch formula, one notices that the χm (X) are determined by (χ(X), χ2 (X), B(X)). Deﬁnition 3.8. A formal basket is a triple B = (χ, χ2 , B), where χ, χ2 are integers and B is a basket.

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Given a formal basket B, one can deﬁne its volume K 3 (B) and the Euler characteristics χm (B) by formally applying Riemann-Roch. We have the following properties which justify the notion of packing and canonical sequences. Proposition 3.9. Given B = (χ, χ2 , B) and BD = (χ, χ2 , B D ) such that B e B D is a packing. Then (1) K 3 (B) > K 3 (BD ); (2) χm (B) ≥ χm (BD ) for all m ≥ 2 and strict inequality holds for some m. Proposition 3.10. Given B = (χ, χ2 , B). Then (1) χm (B) = χm (B (0) (B)) for all 2 ≤ m ≤ 5; (2) χ6 (B) = χ6 (B (0) (B)) − r5 (B); (3) χm (B) = χm (B (n) (B)) for all 2 ≤ m ≤ n + 1 and n ≥ 5; (4) χn+2 (B) = χn+2 (B (n) (B)) − rn+1 (B) for all n ≥ 5. We have the following rough ﬁniteness principle which is useful in various situations. However, one needs to work out reﬁnements in various applications. Proposition 3.11. Suppose that χ, χ2 , χ3 are bounded below and bounded above. Suppose furthermore that R := max{ri |(bi , ri ) ∈ B} is bounded above. Then there are only ﬁnitely many formal baskets with the given conditions. Proof. By the Riemann-Roch formula, we have M σ(B) := bi = 10χ + 5χ2 − χ3 . (bi ,ri )∈B

This gives the boundedness of σ(B). Together with the boundedness of R, we have the boundedness of baskets, hence of formal baskets.

4. Weak Q-Fano threefolds A threefold X is said to be a terminal (resp. canonical) Q-Fano threefold if X has, at worst, terminal (resp. canonical) singularities and −KX is ample, where KX is a canonical Weil divisor on X. A threefold X is called a terminal weak Q-Fano threefold if X has, at worst, terminal singularities and −KX is nef and big. In [29, Section 4.3], Reid conjectured that P−2 (X) > 0 for almost all Q-Fano threefolds. There are already several known examples with P−2 = 0 by IanoFletcher [15] and Altinok and Reid [1]. Another question that we are interested in, is the boundedness of Q-Fano threefolds, which is equivalent to the boundedness of 3 the anti-canonical volume −KX . Kawamata [21] ﬁrst showed the boundedness of 3 −KX for terminal Q-Fano threefolds with Picard number 1. Koll´ar, Miyaoka, Mori and Takagi [27] then gave the boundedness for all canonical Q-Fano threefolds. In [6], we prove the following:

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Theorem 4.1. Let X be a terminal weak Q-Fano threefold. Then (1) P−4 > 0 with possibly one exception of a basket of singularities; (2) P−6 > 0 and P−8 > 1; 1 1 3 3 = 330 ≥ 330 . Furthermore −KX if and only if the virtual basket of (3) −KX singularities is 1 1 1 1 B(X) = { (1, −1, 1), (1, −1, 2), (1, −1, 1), (1, −1, 2)}. 2 5 3 11

Sketch of the proof. We ﬁrst claim that there are only ﬁnitely many formal baskets with P−1 = P−2 = 0. Recall that we have vanishing higher cohomologies, hence P−m = χ−m = −χm+1 , for all m ≥ 1. By [27], one has that −KX · c2 (X) ≥ 0. Therefore, [29, 10.3] gives the following inequality: t t M 1 M ri + 24 ≥ 0. (2.1) − r i=1 i i=1 Therefore, we have boundedness of formal baskets by Proposition 3.11. We can give a complete list of those formal baskets with P−1 = P−2 = 0. It turns out that there is only one such, say B0 . Hence if P−4 (X) = 0, then P−1 (X) = P−2 (X) = 0. So this could only happen if B(X) = B0 . If P−6 (X) = 0, then again P−1 (X) = P−2 (X) = 0. One can get an immediate contradiction for P−6 (B0 ) > 0. A similar method can be applied to study formal baskets with small antiplurigenera. One may need the following geometric constraints to shorten the list 3 of formal baskets. First of all, −K 3 (B) = −KX > 0 gives the inequality: 2 M b i σ D (B) := < 2χ−1 + σ(B) − 6. ri (bi ,ri )∈B

Moreover, by [26, Lemma 15.6.2], whenever P−m > 0 and P−n > 0, one has P−m−n ≥ P−m + P−n − 1. Once we have classiﬁed formal baskets with small anti-plurigenera, it is easy to calculate other anti-plurigenera of a given formal basket on the list. We would like to remark that it is not diﬃcult to see that −K 3 is large as long as the leading anti-plurigenera are large. Therefore, our method also provides an 1 3 . eﬀective way to get a lower bound if −KX ≥ 330 The lower bound

1 330

is optimal due to the following example by Iano-Fletcher:

Example 4.2. ([15, page 158]) The general hypersurface X66 ⊂ P(1, 5, 6, 22, 33) 3 has −KX =

1 330 .

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5. Threefolds of general type Let Y be a nonsingular projective variety of dimension n. It is said to be of general type if the pluricanonical map ϕm := Φ|mKY | corresponding to the linear system |mKY | is birational into a projective space for m D 0. Thus it is natural and important to ﬁnd a constant c(n), so that ϕm is birational onto its image for all m ≥ c(n) and for all Y with dim Y = n. When dim Y = 1, it was classically known that |mKY | gives an embedding of Y into a projective space if m ≥ 3. When dim Y = 2, Kodaira-Bombieri’s theorem [2] says that |mKY | gives a birational map onto the image for m ≥ 5. A natural approach for studying this problem in higher dimensions is an induction on the dimension by using vanishing theorems. This amounts to estimating the plurigenera and for this the greatest diﬃculty seems to be to bound from below the canonical volume Vol(Y ) := lim sup{ m∈Z+

n! dimC H 0 (Y, OY (mKY ))}. mn

The volume is an integer when dim Y ≤ 2. However it is only a rational number in dimension 3 or higher. In fact, it is almost an equivalent question to study the lower bound of the canonical volume. A recent result of Hacon and Mc Kernan [10], Takayama [30] and Tsuji [31] shows the existence of both c(n) and a lower bound of the canonical volume, however non-explicitly. Our main result in [4] and [5] is the following: Theorem 5.1. Let Y be a nonsingular projective threefold of general type. Then we have (1) P12 > 0; (2) P24 ≥ 2. Sketch. Let X be a minimal model of Y . Suppose that P12 (Y ) = 0, then clearly pg (X) = P2 (X) = P3 (X) = P4 (X) = P6 (X) = 0. It is not that immediate as we did in Section 4. However, by those inequalities rn (B(X)) ≥ 0, one ends up with a contradiction. Please see [4, Theorem 4.10] for the details. The proof for P24 ≥ 2 consists of similar but more involved arguments. Once one has Pm0 ≥ 2, the m0 -canonical produces a non-trivial ﬁbration. Indeed, Meng Chen [9, Theorem 0.1] improved Koll´ ar’s result in [25, Corollary 4.8] to obtain the following: Theorem 5.2. Let Y be a nonsingular projective threefold of general type with Pm0 ≥ 2. Then ϕm is birational for all m ≥ 5m0 + 6. Therefore, we have that ϕm is birational onto its image for all m ≥ 126 and 1 Vol(Y ) ≥ 63·126 2 for a threefold of general type. We improve the bound by a more detailed study of various types of ﬁbrations in [5]. We have

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Theorem 5.3. Let Y be a nonsingular projective threefold of general type. Then one has (1) ϕm is birational for m ≥ 73; (2) Vol(Y ) ≥

1 2660 ;

(3) Vol(Y ) ≥

1 420

when χ(OX ) = 1.

Sketch. The main idea of the proof of the above results is as follows. If χ(O) ≤ 0, then we know that ϕ5 is birational by [7]. We thus assume that χ(O) > 0. If any of Pm0 ≥ 2 for some 2 ≤ m0 ≤ 12, then ϕm is birational for all m ≥ 66 by [9, Theorem 0.1]. We hence assume that Pm0 ≤ 1 for all 2 ≤ m0 ≤ 12. We assume that X is minimal, hence Pm = χm for m ≥ 2 thanks to the vanishing theorem. By the inequality (3.1), one sees that χ ≤ 8. Hence it remains to consider threefolds of general type with 1 ≤ χ ≤ 8 and χm = 0, 1 for 2 ≤ m ≤ 12. The case with χ(OX ) = 1 has been studied by Fletcher intensively. We classify formal baskets with 2 ≤ χ ≤ 8 and χm = 0, 1 for 2 ≤ m ≤ 12 in [5]. Example 5.4. (see [15, p. 151, No.23]) The “worst” known example is a general weighted hypersurface X = X46 ⊂ P(4, 5, 6, 7, 23). The threefold X has invariants: pg (X) = P2 (X) = P3 (X) = 0, P4 (X) = · · · = P9 (X) = 1, P10 (X) = 2 and 1 Vol(X) = 420 . Moreover, it is known that ϕm is birational for all m ≥ 27, but ϕ26 is not birational.

6. Weighted complete intersections As one may have noticed, the worst known examples are certain hypersurfaces in weighted projective spaces. An immediate expectation is that one might be able to produce many more examples by considering weighted complete intersections. In [3], we prove that there is no new example with terminal singularities other than those on Fletcher’s list [15]. Given a weighted complete intersection % % X = Xd1 ,...,dc ⊂ P(a0 , . . . , an ), its amplitude is deﬁned to be α := j dj − i ai , so that ωX ∼ = OX (α). We ﬁrst prove that the codimension can not be arbitrarily large. Theorem 6.1. Let X = Xd1 ,...,dc ⊂ P(a0 , . . . , an ) be a quasi-smooth weighted complete intersection of amplitude α and codimension c that is not an intersection with a linear cone. Then 6 dim X + α + 1 if α ≥ 0, c≤ dim X if α < 0. The proof consists of a careful computation of the rank of the Jacobian matrix of the deﬁning equations. As a consequence, we prove that Fletcher’s lists of quasi-smooth weighted complete intersection with only terminal singularities [15, 15.1, 15.4, 18.16] are complete. To provide more evidence for this, let us consider the case that α = 1 and

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ai > 6 for all i. Then we have Pm (X) = 0 for all m ≤ 6. This leads to a contradiction as in the proof of Theorem 5.1. Therefore, we have at least a0 ≤ 6. In fact, by counting the number of {ai |ai ≤ 6} and {dj |dj ≤ 6}, one can determine P2 , . . . , P6 . Then one can determine B (0) (X). Notice that there are only ﬁnitely many baskets with given B (0) . Hence, we get ﬁnitely many formal baskets. For each formal basket, we can compute its Poincar´e series and run the “table method” (cf. [15, Section 18]) to determine whether the given formal basket can be realized by a weighted complete intersection or not.

7. Divisorial contraction to a point The purpose of this section is to review some results on divisorial contractions in dimension three. We distinguish divisorial contractions into the following three types. The ﬁrst one is divisorial contractions to a point of index 1. The second is divisorial contractions to a point of index r > 1 with discrepancy 1r . The third is divisorial contractions to a point of index r with discrepancy > 1r . In [16], Kawakita showed that a divisorial contraction to a smooth point is a weighted blow-up with weight (1, a, b) for some (a, b) = 1. He then proved in [17] and [18] that any divisorial contraction to a cA point can be described as a weighted blow-up except for one case, which is a contraction to a cA2 point with discrepancy 3. For the second case of divisorial contraction to a point of index r > 1 with discrepancy 1r , Hayakawa shows that any such contraction is a weighted blow-up. We would like to remark that one might need to change the embedding of the singular point in order to see the description. There are even a few cases where one needs to reembed the singular point into C5 /µr (cf. [11], [12]). We summarize his results together with Kawamata’s and Kawakita’s in the following: Theorem 7.1. Let (P ∈ X) be a point of index r > 1. (1) There is a weighted blow up Y → X at (P ∈ X) with discrepancy 1/r. (2) Extremal divisorial contractions to (P ∈ X) with discrepancy 1/r are given by the weighted blow-ups classiﬁed in [11], [12]. (3) Extremal divisorial contractions to (P ∈ X) with discrepancy greater than 1/r are classiﬁed in [19]. (4) There is a partial resolution Xn → Xn−1 → . . . → X0 = X, such that Xn has Gorenstein singularities and each map is a weighted blow-up ˜X /EX ) over a singular point of index ri > 1 as in (1). We have that n ≤ ρ(E ˜ where EX ∈ | − KX | is general and EX → EX is the minimal resolution. Proof. See [22], [11], [12] and [13].

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Deﬁnition 7.2. A w-morphism will denote an extremal divisorial contraction to a point (P ∈ X) of index r > 1 with discrepancy 1/r (see 7.1). A partial resolution as in (4) of 7.1 will be called a w-resolution of (P ∈ X). The remaining case is divisorial contractions to a point of index r > 1 with discrepancy > 1r . We consider these as exceptional cases. Kawakita gave a complete description of these contractions, which can be classiﬁed into 6 types. Hayakawa shows that any such contraction can be described by a weighted blow-up except one case with discrepancy 2 over a cD/2 point. Please see [14] for details. Proposition 7.3. Let f : Y → X be a divisorial contraction to a point of index 1 or a w-morphism. Then we have (1) K 3 (Y ) > K 3 (X), (2) χm (Y ) ≥ χm (X) for all m ≥ 2 and strict inequality holds for some m. Proof. Let E be the f -exceptional divisor. Since KY = f ∗ KX + aE for some a > 0 and E 3 > 0, the ﬁrst inequality is clear. To see the second inequality, we use the singular Riemann-Roch formula. We have m(m − 1)(2m − 1) 3 3 χm (Y ) − χm (X) = a E 12 m−1 m−1 M M M M + l(j, Q) − l(j, P ). j=1 Q∈B(Y )

j=1 P ∈B(X)

In the case that P := f (E) ∈ X is a point of index 1, then clearly f (E) produces no basket; together with the isomorphism Y − E → X − f (E), we see the inequality. It is in fact a strict inequality for all m ≥ 2. If f is a divisorial contraction to a point of index r > 1 and discrepancy 1r , then one can verify that χ2 (Y ) = χ2 (X) and B(Y ) e B(X) by the classiﬁcation of Hayakawa. Therefore, we have B(Y ) e B(X). Hence the inequalities follow from Proposition 3.9.

8. Depth of singularities We would like to introduce a notion called depth in order to measure the complexity of singularities. Recall that there is a well-known notion of diﬃculty which is used to prove the termination of ﬂips. However, the behavior of diﬃculty might be diﬀerent from the usual expectation in certain divisorial contractions. Deﬁnition 8.1. Let (P ∈ X) be a terminal singularity, and let f : Y → X be a resolution. Let d(P ∈ X) := #{Ei |a(X, Ei ) < 1, f (Ei ) = P } be the number of exceptional divisors with discrepancy < 1, which is called the diﬃculty.

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Note that d(P ∈ X) = 0 if P ∈ X is Gorenstein. Therefore, it is non-zero only at those % points of index > 1, which are ﬁnite in number. It makes sense to deﬁne d(X) := P ∈X d(P ∈ X). The following properties of diﬃculty among elementary birational maps are easy. Proposition 8.2. (1) If Y → X is a divisorial contraction to a point, then d(Y ) + 1 ≥ d(X). (2) If Y → X is a divisorial contraction to a curve, then d(Y ) ≥ d(X). (3) If X ''( X + is a ﬂip, then d(X) > d(X + ). Deﬁnition 8.3. We deﬁne the depth of (P ∈ X), denoted dep(P ∈ X), to be the minimum length of all w-resolutions of (P ∈ X). Note that dep(P ∈ X) = 0 if and only if (P ∈ X) is Gorenstein. Therefore, it is non-zero only at the points % of index > 1, which are ﬁnite in number. It makes sense to deﬁne dep(X) := P ∈X dep(P ∈ X). Example 8.4. Let (P ∈ X) be a cyclic point of index r, then there is an economic resolution f : Y → X such that KY = f ∗ KX +

r−1 M i Ei . r i=1

We thus have d(P ∈ X) = r − 1. The only w-morphisms are given by weighted blow ups f : Y → X such that Y contains (at most) two quotient singularities of indices a and r − a. The proof now follows easily by induction on the index. This also reproduces the economic resolution. We thus have dep(P ∈ X) = r − 1. Example 8.5. The notion of depth is similar to but diﬀerent from the usual notion of diﬃculty d(X). For example, take a cA/2 point (P ∈ X) deﬁned by (xy + z 6 + u3 = 0)/ 21 (1, 1, 1, 0). We consider a w-morphism f : Y → X which is a weighted blow-up with weights 21 (5, 1, 1, 2). One has KY = f ∗ KX + 12 E and there is only a cyclic quotient singularity Q1 on Y of type 51 (2, −1, −2) which is equivalent to 15 (3, 1, 2). Let g : Z → Y be the economic resolution of Q1 ∈ Y such that Mi KZ = g ∗ KY + Ei . 5 i Also g ∗ E = EZ +

M 3i { }Ei , 5 i

where EZ denotes the proper transform of E in Z and {x} denotes the fractional part of x ∈ Q. We thus have 1 1 1 K Z = g ∗ f ∗ K X + EZ + E1 + E2 + E3 + E4 . 2 2 2

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Thus

d(Y ) = 4,

while

dep(Y ) = 4,

d(X) = 3, dep(X) = 5.

Proposition 8.6. We have the inequality dep(X) ≥ d(X). Proof. It suﬃces to show dep(P ∈ X) ≥ d(P ∈ X) for all P ∈ X. Let g : Xn → Xn−1 → . . . → X0 = X be a w-resolution which realizes the depth of P ∈ X. Let f : Z → Xn be a resolution of possible Gorenstein points on Xn . Let Ei be the exceptional divisor of the w-morphism τi−1 : Xi → Xi−1 and E˜i be its proper transform in Z. Let fi : Z → Xi be the corresponding map to Xi . We have M KZ = f ∗ KXn + b k Fk , k

with bk ≥ 1 ∈ Z and

1 Ei , ri M M aij E˜j + fi∗ Ei = E˜i + bik Fk , ∗ (KXi−1 ) + KXi = τi−1

j>i

k

for some aij , bik ≥ 0 ∈ Q. In total, we have KZ = f0∗ KX +

n M M M M 1 1 1 aij )E˜j + bik )Fk . (bk + ( + r r r i i i<j i j=1 j k

It is clear that the number of divisors with discrepancy < 1, which is the diﬃculty d(X), is less than or equal to n = dep(X). Example 8.7. We ﬁrst consider the case in [19, 1.2.ii.a], where f : Y → X is a divisorial contraction to a point P ∈ X of type cD/2 with discrepancy a/2 > 1/2. The corresponding divisorial contraction is given by a weighted blow-up Y → X with weights σa = 21 (a, s, 2, s + 2). We have that a|(s + 1), a ≡ s ≡ 1 (mod 2), and so we may assume that s + 1 = 2ad. We may also assume that a ≥ 3. Let {Ui }i=1,...,4 be the standard covering of the weighted blow-up and let Qi be the origin of Ui . One sees that Q2 (resp. Q4 ) is a cyclic quotient point of index s (resp. s + 2). Q3 is a terminal singularity of type cD/2. One can construct a sequence of varieties Za → Za−1 → . . . Z0 = X inductively by weighted blow-ups such that (1) On Zk , there is a cD/2 singularity Rk3 for k ≥ 1. There is a cD/2 singularity (P ∈ Z0 ). (2) Zk+1 → Zk is the weighted blow-up over the point Rk3 (over the point P when k = 0) with weights 12 (1, 2d − 1, 2, 2d + 1) (resp. 21 (1, 2d + 1, 2, 2d − 1)) if k is even (resp. odd).

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(3) On Zk , there are two cyclic singularities Rk+ , Rk− of index 2d + 1, 2d − 1 respectively such that {Rk+ , Rk− } = {Rk2 , Rk4 }. (4) (Ra3 ∈ Za ) ∼ = (Q3 ∈ Y ). (5) Zk is terminal and φk : Zk+1 → Zk is a w-morphism for all k. We then have that

%a dep(X) ≤ a + dep(Ra3 ) + k=1 (dep(Rk+ ) + dep(Rk− )) = a + dep(Q3 ) + a(2d − 2 + 2d) = dep(Q3 ) + 2r + 2 − a = dep(Y ) + 2 − a.

The other exceptional divisorial contractions to a point have a similar description as long as these can be realized as a weighted blow-up. By [14], except for the divisorial contraction with discrepancy 2 over a cD/2 point, one can realize it as a weighted blow-up and compute the depth explicitly as in the above example. The comparison of the remaining case with discrepancy 2 over a cD/2 point can be carried out by using cohomological methods and the singular Riemann-Roch formula. We thus have the following properties. Proposition 8.8. Let f : Y → X be a divisorial contraction to a point. Then dep(Y ) + 1 = dep(X) only when f is a w-morphism. Proposition 8.9. (1) If Y → X is a divisorial contraction to a point, then dep(Y ) + 1 ≥ dep(X). (2) If Y → X is a divisorial contraction to a curve, then dep(Y ) ≥ dep(X). (3) If X ''( X + is a ﬂip, then dep(X) > dep(X + ).

9. Factorization of birational maps The purpose of this section is to show that one can factorize a birational map in dimension three into more elementary ones. By using the classiﬁcation of extremal neighborhood and weighted blow-up over the point of highest index, we prove the following: Theorem 9.1. Let g : X ⊃ C → W _ R be an extremal neighborhood. If X is not Gorenstein, then for any w-morphism f : Y → X over the point of highest index, we have CY · KY ≤ 0, where CY denotes the proper transform of C in Y . We would like to remark that Kawamata proved this kind of fact for the semistable case in [23]. Notice also that this can not hold for an arbitrary divisorial contraction to a point. For example, consider the weighted blow-up Y → X with discrepancy a2 as in Example 8.7. One sees that a KY · CY = KX · CX + E · CY > 0, 2

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by picking a D 0. By using Theorem 9.1, one has either a ﬂip or a ﬂop Y ''( Y + . Running the minimal model program of Y over W , we obtain the following: Theorem 9.2. Let g : X ⊃ C → W _ P be an extremal neighborhood which is isolated (resp. divisorial). If X is not Gorenstein, then we have a diagram where Y ''( Y D consists of ﬂips and ﬂops over W , f : Y → X is a w-morphism, f D : Y D → X D is a divisorial contraction (resp. a divisorial contraction to a curve) and g D : X D ∼ = X + → W is the ﬂip of g (resp. g D is a divisorial contraction to a point). By induction on the depth of X, we are able to prove the following factorization result together with some interesting properties of the notion of depth. Theorem 9.3. Let g : X → W be a ﬂipping contraction and let φ : X''(X + be the corresponding ﬂip, then φ can be factored as f0

fn

X = X0 ''( X1 ''( . . . ''( Xn ''( X + , such that each fi is the inverse of one of the following: a w-morphism, a ﬂop, a blow-down to an LCI curve or a divisorial contraction to a point. Let g : X → W be a divisorial contraction to a curve, then g can be factored as f0

fn

X = X0 ''( X1 ''( . . . ''(Xn ''( W, such that each fi is the inverse of one of the following: a w-morphism, a ﬂop, a blow-down to an LCI curve or a divisorial contraction to a point. The idea is that instead of using all divisorial contractions to a point, we use only w-morphisms to resolve singularities on an extremal neighborhood C ⊂ X. By Theorem 9.1, one can run the minimal model program in the category of varieties with milder singularities measured by the depth dep(X).

References [1] S. Altinok, M. Reid, Three Fano threefolds with | − K| = ∅, preprint. [2] E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Etudes Sci. Publ. Math. 42 (1973), 171–219. [3] J.J. Chen, J.A. Chen, M. Chen, On weighted complete intersections, Jour. Alg. Geom, to appear. arXiv 0908.1439. [4] J.A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, I, Ann. Sci. Ecole Norm. Sup. 43 (2010), 365-394. [5] J.A. Chen, M. Chen, Explicit birational geometry of threefolds of general type, II, Jour. Diﬀ. Geom., to appear. arXiv: 0810.5044. [6] J.A. Chen, M. Chen, An optimal boundedness on weak Q-Fano threefolds, Adv. Math., 219, (2008), 2086-2104.

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[7] J.A. Chen, M. Chen, D.-Q. Zhang, The 5-canonical system on 3-folds of general type, J. reine angew. Math. 603 (2007), 165–181. [8] J.A. Chen, C.D. Hacon, Factoring 3-fold ﬂips and divisorial contractions to curves, preprint, arXiv 0910.4209. [9] M. Chen, On the Q-divisor method and its application, J. Pure Appl. Algebra 191 (2004), 143–156. [10] C.D. Hacon and J. Mc Kernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1-25. [11] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities, Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 515–570. [12] T. Hayakawa, Blowing ups of 3-dimensional terminal singularities. II, Publ. Res. Inst. Math. Sci. 36 (2000), no. 3, 423–456. [13] T. Hayakawa, Gorenstein resolutions of 3-dimensional terminal singularities, Nagoya Math. J. 178 (2005), 63–115. [14] T. Hayakawa, Divisorial contractions to 3-dimensional terminal singularities with discrepancy one, J. Math. Soc. Japan 57 (2005). [15] A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of threefolds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. [16] M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math. 145 (2001), no. 1, 105–119. [17] M. Kawakita, Divisorial contractions in dimension three which contract divisors to compound A1 points, Compositio Math. 133 (2002), no. 1, 95–116. [18] M. Kawakita, General elephants of three-fold divisorial contractions. J. Amer. Math. Soc. 16 (2003), no. 2, 331–362. [19] M. Kawakita, Three-fold divisorial contractions to singularities of higher indices, Duke Math. J. 130 (2005), no. 1, 57–126. [20] Y. Kawamata, On the plurigenera of minimal algebraic 3-folds with K ≡ 0, Math. Ann. 275 (1986), no. 4, 539–546. [21] Y. Kawamata, Boundedness of Q-Fano threefolds, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), 439–445, Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence, RI, 1992. [22] Y. Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento, 1994), 241–246, de Gruyter, Berlin, 1996. [23] Y. Kawamata, Semistable minimal models of threefolds in positive or mixed characteristic, J. Algebraic Geom. 3 (1994), no. 3, 463–491. [24] K. Kodaira, On compact complex analytic surfaces. I, Ann. of Math. (2) 71 (1960), 111–152; II, ibid. 77 (1963), 563–626; III ibid. 78 (1963), 1–40. [25] J. Koll´ ar, Higher direct images of dualizing sheaves I, Ann. Math. 123 (1986), 11-42; II, ibid. 124 (1986), 171-202. [26] J. Koll´ ar, Shafarevich maps and automorphic forms, M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1995.

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[27] J. Koll´ ar, Y. Miyaoka, S. Mori, H. Takagi, Boundedness of canonical Q-Fano threefolds, Proc. Japan Acad. 76, Ser. A (2000), 73-77. [28] S. Mori, On 3-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66. [29] M. Reid, Young person’s guide to canonical singularities, Proc. Symposia in Pure Math. 46 (1987), 345-414. [30] S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551–587. [31] H. Tsuji, Pluricanonical systems of projective varieties of general type, I, Osaka J. Math. 43 (2006), 967–995. Jungkai Alfred Chen, Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan E-mail: [email protected]

Emptiness of homogeneous linear systems with ten general base points Ciro Ciliberto, Olivia Dumitrescu, Rick Miranda, Joaquim Ro´e

Abstract. In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.

Introduction We will denote by Ld (ms11 , ..., msnn ) the linear system of plane curves of degree d having multiplicities at least mi at si ﬁxed points, i = 1, . . . , n. The points in question may be proper or inﬁnitely near, but often we will assume them to be general. In the homogeneous case, the linear system Ld (mn ) has the expected dimension d(d + 3) nm(m + 1) − }. e(Ld (mn )) = max{−1, 2 2 √ d Nagata’s conjecture for ten general points states that if m < 10 ≈ 3.1622 then d Ld (m10 ) is empty. Harbourne and Ro´e [7] proved that if m < 177/56 ≈ 3.1607 then Ld (m10 ) is empty. Then Dumnicki [5] (see also [1]), combining various techniques, among which methods developed by Ciliberto–Miranda [2] and Harbourne–Ro´e, found a better bound 313/99 ≈ 3.161616. The aim of this paper is to develop a general degeneration technique for analysing the emptiness of Ld (mn ) for general points, and we demonstrate it here in the case n = 10. This technique is based on the blow–up and twist method introduced in this setting by Ciliberto and Miranda in [2]. Using this, and precisely exploiting a suitable degeneration of the plane blown up at ten general points into a union of nine surfaces, we prove that Ld (m10 ) d < 117/37 ≈ 3.162162. Using the same degeneration Ciliberto and is empty if m d Miranda recently proved in [4] the non-speciality of Ld (m10 ) for m ≥ 174/55 and, as remarked in that article, one obtains as a consequence the emptyness of d Ld (m10 ) for m < 550/174 ≈ 3.1609. Our emptiness result implies that the 10– point Seshadri constant of the plane is at least 117/370. In fact, in [7] the authors show that this holds if certain systems of the form Ld (m9 , m + k), with k ≥ 0 and d m < 117/37 are empty (see Table 1 of [7]). Their emptiness is implied by our result. Recently T. Eckl [6] also obtained the same bound. Using the methods developed in [4] he constructs a more complicated degeneration of the plane into 17 surfaces to ﬁnd the bound 370/117 for asymptotic non–speciality of Ld (m10 ). As proved in [4] this is equivalent to saying that the Seshadri constant has to be at least 117/370, which is the same conclusion we obtain here with considerably less eﬀort.

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Ciro Ciliberto, Olivia Dumitrescu, Rick Miranda, Joaquim Ro´e

The present paper has to be considered as a continuation of [4], which the interested reader is encouraged to consult for details on which we do not dwell here. From [4] we will take the general setting and most of the notation. Indeed, the degeneration we use here has been introduced in [4], §9. It is a family parametrized by a disk whose general member Xt is a plane blown up at ten general points, whereas the central ﬁbre X0 is a local normal crossings union of nine surfaces. This construction is brieﬂy reviewed in §1. A line bundle on X0 is the datum of a line bundle on the normalization of each component, verifying matching conditions, i.e. the line bundles have to agree on the double curves of X0 . Such a line bundle is a limit of Ld (m10 ) if there is a line bundle L on the total space which restricts to Ld (m10 ) on the general ﬁbre and restricts to the given bundle on X0 . This implies some additional numerical conditions for the bundles on each component. In order to analyse the emptiness of Ld (m10 ) in the asserted range, we use the concept of central eﬀectivity introduced in [4], §10.1. A line bundle L0 on X0 is centrally eﬀective if a general section of L0 does not vanish identically on any irreducible component of X0 . In particular, if L0 is centrally eﬀective then its restriction to each component of X0 is eﬀective. If Ld (m10 ) is not empty, then there is a line bundle L on the total space X of the family with a non–zero section s vanishing on a surface whose restriction to the general ﬁber Xt is a curve in Ld (m10 ). Then there is a limit curve in the central ﬁber X0 as well, hence there is a limit line bundle L0 associated to that curve. The bundle L0 , which is the restriction to X0 of L twisted by multiples of the components of X0 where s vanishes, is centrally eﬀective. In conclusion, if Ld (m10 ) = K ∅ then there is a limit line bundle which is centrally eﬀective. Conversely if for ﬁxed d and m no limit line bundle L0 is centrally eﬀective, e.g. if its restriction to some component of X0 is not eﬀective, then we conclude that Ld (m10 ) = ∅. In this article we will exploit this argument. We will describe in §2 all limit line bundles L0 of the line bundle Ld (m10 ). We will see that, in order to apply the central eﬀectivity argument, we can restrict our attention to some extremal limit line bundles, and verify central eﬀectivity properties only for them. In §3 we will d < 117/37, by showing prove that Ld (m10 ) with general base points is empty if m that none of the extremal limit line bundles veriﬁes the required central eﬀectivity properties.

1. The degeneration Consider X → Δ the family obtained by blowing up a point in the central ﬁber of the trivial family over a disc Δ × P2 → Δ. The general ﬁbre Xt for t K= 0 is a P2 , and the central ﬁbre X0 is the union of two surfaces P ∪ F, where P ∼ = P2 , F ∼ = F1 , and P and F meet along a rational curve E which is the (−1)–curve on F and a line on P (see Figure 1 in [4]). Choose four general points on F and six general points on P. Consider these as limits of ten general points in the general ﬁbre Xt and blow them up in the family X (we abuse notation and denote by X also the new family). This creates

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191

ten exceptional surfaces whose intersection with each ﬁber Xt is a (−1)–curve, the exceptional curve for the blow–up of that point. The general ﬁbre Xt of the new family is a plane blown up at ten general points. The central ﬁbre X0 is the union of V1 , a plane blown up at four general points, and Z1 , a plane blown up at seven general points (see Figure 2 in [4]). This is the ﬁrst degeneration in [4], §3. We will brieﬂy recall the notion of a 2–throw as described in [4], §4.2. Consider a degeneration of surfaces containing two components V and Z, transversely meeting along a double curve R. Let E be a (−1)–curve on V intersecting R transversely twice. Blow it up in the total space. This creates a ruled surface T ∼ = F1 meeting V along E; the double curve V ∩ T is the negative section of T . The surface Z is blown up twice, with two exceptional divisors G1 and G2 . Now blow up E again, creating a double surface S ∼ = F0 in the central ﬁbre meeting V along E and T along the negative section. The blow–up aﬀects Z, by creating two more exceptional divisors F1 and F2 which are (−1) curves, while G1 and G2 become (−2)–curves. Blowing S down by the other ruling contracts E on the surface V ; R becomes a nodal curve, and T changes into a plane P2 (see Figure 3 in [4]). In this process Z becomes non–normal, since we glue F1 and F2 . However, in order to analyse divisors and line bundles on the resulting surface we will always refer to its normalization Z. On Z we introduced two pairs of inﬁnitely near points pi , qi , corresponding to the (−1)–cycles Fi + Gi and Fi , i = 1, 2. Given a linear system L on Z, denote by L also its pull–back on the blow–up and consider the linear system L(−a(Fi + Gi ) − bFi ). We will say that this system is obtained by imposing to L a point of type [a, b] at pi , qi . The above discussion is general; we now apply it to the degeneration V1 ∪ Z1 described above. Perform the sequence of 2–throws exploiting the following (−1)– curves: (1) The cubic L3 (2, 16 ) on Z1 . This creates the second degeneration in [4], §6 (see Figure 5 there). Note that V1 becomes an 8–fold blow–up of the plane: it started as a 4–fold blow–up and it acquires two more pairs of inﬁnitely near (−1)–curves. (2) Six disjoint curves, i.e. two conics C1 = L2 (14 , [1, 0], [0, 0]), C2 = L2 (14 , [0, 0], [1, 0]) and four quartics Qi = L4 (23 , 1, [1, 1]2 ) on V1 (the multiplicity one proper point is located at the i-th point of the four we blew up on P). Throwing the conics creates the third degeneration in [4], §7 (see Figure 5 there), and further throwing the quartics creates the fourth degeneration in [4], §9 (see Figure 7 there). By executing all these 2–throws we introduce seven new surfaces T , Ui , i = 1, 2 (denoted by T4 , Ui,4 , i = 1, 2 in [4]) and Yj , j = 1, . . . , 4. They are all projective planes, except T , which is however a plane at the second degeneration level. Moreover, we have the proper transforms V and Z of V1 and Z1 (denoted V4 and Z4 in [4]). Throwing the two conics Ci , both Z1 and the plane corresponding to T undergo four blow–ups, two of them inﬁnitely near. By throwing the four

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quartics Qj , V1 becomes more complicated with 16 additional blow–ups, in eight pairs of inﬁnitely near points.

2. The limit line bundles Next we describe in more detail the limit line bundles of Ld (m10 ). Their restrictions to the components of the central ﬁbre will in general be of the form LZ = LdZ (µ, q 6 , [xi , xDi ]i=1,2 ), LT = LdT ([xi , xDi ]i=1,2 ),

LV = LdV (ν 4 , [y, y D ]2 , [zi , ziD ]2i=1,...,4 )

LUi = Lsi , i = 1, 2,

dZ , µ, q, xi , xDi , ...

LYi = Lti , i = 1, . . . , 4

where the parameters etc. are integers. Note that in LZ and LV the points are no longer in general position, since they have to respect constraints dictated by the 2–throws. Our purpose in this section is to describe more precisely the numerical conditions imposed by the matching and the requirement of being a limit. The matching conditions involving the Ui ’s and the Yi ’s, imply si = xi − xDi , i = 1, 2, and ti = zi − ziD , i = 1, . . . , 4. Next we have to impose the remaining matching conditions and also the conditions that this is a limit line bundle of Ld (m10 ), i.e. conditions telling us that the total degree of the limit bundle is d and the multiplicity at the original blown up points is m. This would give us the form of all possible limits line bundles of Ld (m10 ), that we need in order to apply the central eﬀectivity argument. However we can simplify our task, by making the following remark. Let us go back to the 2–throw construction. Let L be an eﬀective line bundle on the total space of the original degeneration such that L · E = −σ < 0. Assume σ = 2h is even. This will be no restriction in our setting, because if we want to prove that a system is empty, it suﬃces to prove that the double is empty. Create the two exceptional surfaces S and T and still denote by L the pull–back of the line bundle on the new total space. In order to make it centrally eﬀective we have to twist it to L(−uT − vS), and central eﬀectivity requires u ≥ h, u ≥ v ≥ 0 and u + v ≥ 2h (see [3], §2). The main remark is that in our setting we may assume u + v = 2h by replacing (u, v) with (uD , v D ) where uD = min{u, 2h}, v D = 2h − uD . Indeed, u+v > 2h means subtracting E more than 2h times from LV , and creating points of type [u, v] rather than [uD , v D ] for LZ . In both cases, this imposes more conditions on the two systems. This is clear for LV . As for LZ , this follows from u(Fi +Gi )+vFi ≥ uD (Fi +Gi )+v D Fi , i = 1, 2. Therefore if one is able to prove that either one of the two systems on V and Z is empty, the central eﬀectivity argument will certainly apply to the original twist L(−uT − vS). Note that u + v = 2h is equivalent to requiring that L(−uT − vS) · E = 0. Essentially the same argument shows that we can also assume that (u, v) = (h, h). The above discussion shows that, in particular, we may assume xi = xDi , i = 1, 2, y = y D , and zi = ziD , i = 1, . . . , 4, with the further conditions that the restrictions to the 2–thrown curves have degree 0. We call extremal the bundles verifying these conditions. If, for given d and m, for all extremal limit line bundles either LZ or

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Emptiness of homogeneous linear systems with ten general base points

LV are empty, then there is no centrally eﬀective limit line bundle and therefore Ld (m10 ) is empty for general points. For an extremal bundle, matching between V and T says that dT = 2x1 = 2x2 . So we set x1 = x2 = x. The multiplicity conditions for the general points on V then read M m = ν + 4x + 2zi + 4 zj , i = 1, . . . , 4 j9=i

yielding z1 = . . . = z4 , which we denote by z. Thus we have eight parameters dV , dZ , ν, µ, q, x, y, z subject to the following seven linear equations 3dZ − 2µ − 6q = 2dV − 4ν − y = 4dV − 7ν − 4y = 0 m = ν + 4x + 14z = q + 2x + 16z + 2y,

d = dZ + 6y + 48z + 6x,

dV − 4y = µ − 4x.

The ﬁrst three come from the zero restriction conditions to the 2-thrown curves, the next two from the multiplicity m conditions on V and Z, the next one from the degree d condition, the last from the matching between V and Z. Set α = d − 3m and ] = 19m − 6d. By solving the above linear system, we ﬁnd dZ = 10α − 6a,

µ = 6α − 3a,

dV = 9a − 18],

q = 3α − 2a,

ν = 4a − 8],

3 x = 5m − d − a 2

y = 2a − 4],

z=

] . 2

The solutions, as is natural, depend on a parameter a ∈ Z (which is the one introduced in the ﬁrst degeneration in [4]). They are integers since we may assume d and m to be even. In conclusion we proved: Proposition 2.1. In the above degeneration, the extremal limit line bundles L of Ld (m10 ) with general base points restrict to the components of the central ﬁbre X0 as follows 3 3 LZ = L10α−6a (6α − 3a, (3α − 2a)6 , [5m − d − a, 5m − d − a]2 ) 2 2 ] ] LV = L9a−18\ ((4a − 8])4 , [2a − 4], 2a − 4]]2 , [ , ]8 ) 2 2 3 3 LT = L10m−3d−2a ([5m − d − a, 5m − d − a]2 ), 2 2 LUi = L0 , i = 1, 2,

LYi = L0 , i = 1, . . . , 4.

If for all a ∈ Z either LZ or LV is empty, then no limit line bundle of Ld (m10 ) on X0 is centrally eﬀective, hence Ld (m10 ) is empty.

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Remark 2.2. As in [4], it is convenient to consider Cremona equivalent models of the linear systems LV and LZ appearing in Proposition 2.1. The system LV is Cremona equivalent to La−2\ ([ 2\ , 2\ ]8 ). The position of the eight inﬁnitely near singular points is special: there are two conics Γ1 , Γ2 intersecting at four distinct points (the contraction of the four quartics), and each of them contains four of the inﬁnitely near points. The conics Γ1 , Γ2 are the proper transforms of F1 , F2 . For all this, see [4], Lemma 9.1. The system LZ is Cremona equivalent toL76d−240m−3a ((13d−41m−a)6 , ( 69 2 d− 109m − a)4 ). This reduction follows by Lemma 9.2 of [4], but one has to apply a further quadratic transformation based at the three points of multiplicity α − ] − a of the system there.

3. Proof of the theorem We can now prove our result: Theorem 3.1. If points is empty.

d m

<

117 37

then the linear system Ld (m10 ) with ten general base

Proof. Fix d, m and assume Ld (m10 ) K= ∅. According to Proposition 2.1, there is an integer a such that both LV and LZ are not empty. Look at the system LV , or rather at its Cremona equivalent form La−2\ ([ 2\ , 2\ ]8 ) (see Remark 2.2). Consider the curve Γ = Γ1 + Γ2 , i.e. the union of the two conics on which the inﬁnitely near base points are located. Blow up these base points. By abusing notation we still denote by Γ and LV the proper transform of curve and system. Then Γ is a 1–connected curve and Γ2 = 0. Since LV is eﬀective, one has LV · Γ ≥ 0, i.e. a ≥ 4]. Consider then LZ , with its Cremona equivalent form L76d−240m−3a ((13d − 4 41m−a)6 , ( 69 2 d−109m−1) ). Since this is eﬀective, we have 76d−240m ≥ 3a ≥ 12], d yielding m ≥ 117 37 .

References [1] T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. Leopold Knutsen, W. Syzdek, T. Szemberg, A primer on Seshadri constants, arXiv:0810.0728v1. [2] C. Ciliberto, R. Miranda, Linear Systems of Plane Curves with Base Points of Equal Multiplicity, Trans. Amer. Math. Soc. 352, 4037-4050 (2000). [3] C. Ciliberto, R. Miranda, Matching Conditions for Degenerating Plane Curves and Applications, in Projective Varieties with Unexpected Properties, Proceedings of the Siena Conference, C. Ciliberto, A. V. Geramita, B. Harbourne, R. M. Mir´ o–Roig, K. Ranestad ed., W. de Gruyter, 2005, 177-198. [4] C. Ciliberto, R. Miranda, Homogeneous interpolation on ten points, arXiv: 0812.0032v1, to appear in J. Alg. Geom.

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[5] M. Dumnicki, Regularity and Non-Emptyness of Linear Systems in Pn , arXiv: 0802.0925v1. [6] T. Eckl, Ciliberto-Miranda degenerations of CP2 blown up in 10 points, arXiv: 0907.4425v1. [7] B. Harbourne, J. Ro´e, Computing Multi-Point Seshadri Constants on P2 , arXiv: math/0309064v3. Ciro Ciliberto, Dipartimento di Matematica, II Universit` a di Roma, Italy E-mail: [email protected] Olivia Dumitrescu, Colorado State University, Department of mathematics, College of Natural Sciences, 117 Statistics Building, Fort Collins, CO 80523 E-mail: [email protected] Rick Miranda, Colorado State University, Department of mathematics, College of Natural Sciences, 117 Statistics Building, Fort Collins, CO 80523 E-mail: [email protected] Joaquim Ro´e, Departament de Matem`tiques, Universitat Aut` onoma de Barcelona, Ediﬁci C, Campus de la UAB, 08193 Bellaterra (Cerdanyola del Vall`es) E-mail: [email protected]

Finite generation of adjoint rings after Lazi´ c: an introduction Alessio Corti Contents 1 Introduction

197

2 Natural operations with divisorial rings

204

3 The main construction

207

4 Lifting lemmas

209

5 Restriction of strictly dlt rings

212

1. Introduction This note is an introduction to all the key ideas of Lazi´c’s recent proof of the theorem on the ﬁnite generation of adjoint rings [Laz09]. (The theorem was ﬁrst proved in [BCHM09].) I try to convince you that, despite technical issues that are not yet adequately optimised, nor perhaps fully understood, Lazi´c’s argument is a self-contained and transparent induction on dimension based on lifting lemmas and relying on none of the detailed general results of Mori theory. On the other hand, it is shown in [CL10] that all the fundamental theorems of Mori theory follow easily from the ﬁnite generation statement discussed here: together, these results give a new and more eﬃcient organisation of higher dimensional algebraic geometry. The approach presented here is ultimately inspired by a close reading of the work of Shokurov [Sho03], I mean speciﬁcally his proof of the existence of 3-fold ﬂips. Siu was the ﬁrst to believe in the possibility of a direct proof of ﬁnite generation, and believing that something is possible is, of course, a big part of making it happen. All mathematical detail is taken from [Laz09]; my contribution is merely exegetic. I begin with a few key deﬁnitions leading to the statement of the main result. 1.1. Basic deﬁnitions. Convention 1.1. Throughout this paper, I work with nonsingular projective varieties over the complex numbers. Let V be a (ﬁnite dimensional) real vector space deﬁned over the rationals. By a cone in V I always mean a convex cone, that is a subset C ⊂ V such that 0 ∈ C

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and: t ≥ 0, v ∈ C ⇒ tv ∈ C , v1 , v2 ∈ C ⇒ v1 + v2 ∈ C. A ﬁnite rational cone is a cone C ⊂ V generated by a ﬁnite number of rational vectors. If U ⊂ V is an aﬃne subspace, then I denote by U (R) and U (Q) the sets of real and rational points of U . The end of a proof of a statement, or the absence of a proof, is denoted by Notation 1.2. I denote by R one of Z, Q, R. If X is a normal variety, I denote by DivR X the group of (Weil) divisors on X with coeﬃcients in R, and by Div+ RX the sub-monoid of eﬀective divisors. In this note, I almost always work with actual divisors, not divisors modulo linear equivalence. For instance, when I write KX , I mean that I have chosen a speciﬁc divisor in the canonical class; the choice is made at the beginning and ﬁxed throughout the discussion. Notation 1.3. If X is a normal variety and D ∈ DivR X is a divisor on X, I write: D = Fix D + Mob D, where Fix D and Mob D are the ﬁxed and the mobile part of D. The deﬁnition makes sense even when D is not integral or eﬀective. Indeed the sheaf OX (D) is deﬁned as 7 G R ( Γ U, OX (D) = f ∈ k(X) | divU f + D|U ≥ 0 and then, by deﬁnition: M Mob D = mE E where

7 G mE = − inf{multE f | f ∈ H 0 X, OX (D) }.

In applications D is almost always integral 7 and G eﬀective. If D is not integral, then the deﬁnition says that OX (D) = OX \DL ; if D is integral, then Fix D = Fix |D| is the ﬁxed part of the complete linear system |D|. Throughout this paper, I use without warning the following elementary fact, often called Gordan’s lemma. Lemma 1.4. Let C ⊂ Rr be a ﬁnite rational cone. The monoid Λ = C ∩ Zr is ﬁnitely generated. Deﬁnition 1.5. Let X be a nonsingular projective variety and Λ = C ∩ Zr where C ⊂ Rr is a ﬁnite rational cone. (1) A divisorial ring on X is a Λ-graded ring of the form . 7 G R(X; D) = H 0 X; D(λ) where D : Λ → DivR X λ∈Λ

is a map such that M (λ) = Mob D(λ) is superadditive, i.e., M (λ1 + λ2 ) ≥ M (λ1 ) + M (λ2 ) for all λ1 , λ2 ∈ Λ. The map D : Λ → DivR X is called the characteristic system of the ring. When I wish to emphasise the grading by Λ, I write R(X; Λ) instead of R(X; D).

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(2) A divisorial ring R(X; D) is adjoint if in addition • D : Λ → DivQ X is rational PL, i.e., it is the restriction of a rational piecewise linear map that, abusing notation, I still denote by D : C → DivR X. This means that there is a ﬁnite decomposition C = ∪m i=1 Ci into ﬁnite rational cones Ci ⊂ Rr such that D|Ci is (the restriction of) a rational linear function. • There 7 is a rational G PL function r : C → R+ and for all λ ∈ Λ, D(λ) = r(λ) KX + Δ(λ) with Δ(λ) ≥ 0. (3) An adjoint ring is big if there is an ample Q-divisor A and Δ(λ) = A + B(λ) with all B(λ) ≥ 0. (4) A big adjoint (snc)G 7 %r ring is klt (dlt) if there is a ﬁxed simple normal crossing divisor j=1 Bj ⊂ X such that all Supp B(λ) ⊂ ∪rj=1 Bj and all X, B(λ) are klt (dlt). Remark 1.6. (1) If D : Λ → Div+ Q X is superadditive and rational PL, then it is also concave. (2) If R(X; D) is an adjoint ring then in particular: Δ : C → R+ is homogeneous of degree 0; that is, Δ(tw) = tΔ(w) for all t ∈ R+ , w ∈ C. (3) Note that in the deﬁnition of dlt (klt) %r big adjoint ring, all divisors in sight are contained in a ﬁxed snc divisor j=1 Bj . In this context, a (big) adjoint ring is dlt (klt) if and only if B(λ) =

r M

bj (λ)Bj

j=1

where all 0 ≤ bj (λ) ≤ 1 (< 1) for all λ ∈ Λ, i = j, . . . , r. In this paper we never need the deﬁnitions, results and techniques of the general theory of singularities of pairs. 1.2. The main result. Theorem 1.7 (Theorem A). [BCHM09] Let X be nonsingular projective, and R = R(X; D) a dlt big adjoint ring on X. Assume, in addition, that D : Λ → Div+ Q X, that is, D(λ) ≥ 0 for all λ ∈ Λ. Then R = R(X; D) is ﬁnitely generated. Remark 1.8. The additional assumption can be removed. The statement is written here in the form that best suits the logic of proof described below in section 1.3. Once Theorem A, and Theorems B and C of section 1.3, are proved, then the additional assumption is removed by a straightforward application of Theorem C.

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Corollary 1.9. If X is nonsingular projective of general type, then the canonical ring R(X, KX ) is ﬁnitely generated. Remark 1.10. In fact, by work of Fujino and Mori [FM00], the results here imply the stronger statement that if X is nonsingular, then the canonical ring of X is ﬁnitely generated. I don’t know if Theorem 1.7 can similarly be strengthened: it would be extremely useful–see [CL10]–if it could. Remark 1.11. • This theorem is proved in [BCHM09]. That proof uses all that is known about the minimal model program; in particular, I mention [FA92, Sho03, HM06, Cor07, BCHM09, HM09]. • The proof by Lazi´c is a self-contained induction on dimension based on lifting lemmas [Siu98], [Cor07, Chapter 5], [HM09], etcetera. On the other hand it is shown in [CL10] that Theorem A readily implies all the fundamental theorems of Mori theory. 1.3. The logic of the proof. In [Laz09] Theorem A is proved by a bootstrap induction on dimension together with two other theorems that I state shortly following some preparations. Asymptotic ﬁxed part. I summarise some facts on asymptotic invariants of divisors, mostly following [ELM+ 06]. For a projective normal variety X, I denote by Eﬀ(X; R) ⊂ N 1 (X; R) the cone of (numerical equivalence classes of) eﬀective divisors with coeﬃcients in R, and by Eﬀ(X; R) ⊂ N 1 (X; R) the cone of pseudo-eﬀective divisors, that is, the closure of the cone of eﬀective divisors (it only makes sense to do this with real coeﬃcients). Similarly, I denote by Big(X; R) ⊂ N 1 (X; R) the cone of big divisors; Big(X; R) is the interior of Eﬀ(X; R). If D ∈ DivQ X is a Q-divisor, then the stable base locus of D is the subset L B(D) = Bs|pD| ⊂ X 0
(if |pD| = ∅ for all 0 < p ∈ Z, then I say by convention that B(D) = X). If D ∈ DivQ X is a big Q-divisor, then the asymptotic ﬁxed part of D is the divisor 1 Fix nD ∈ Div+ F(D) = inf R X. 0
D ∈ Eﬀ(X; R)

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where A is an ample divisor (the deﬁnition is independent of the choice of A). A subtle point is that F(−) is continuous on Big(X; R) (because it is convex), but not necessarily on Eﬀ(X; R). The paper [ELM+ 06] promotes the view that certain asymptotic invariants deﬁned on Eﬀ(X; R), for instance the asymptotic ﬁxed part, are reasonably wellbehaved under surprisingly general conditions. The assertions that follow demonstrate how these invariants are much better behaved on the subcone of adjoint divisors. General setup. In what % follows, X is a nonsingular projective variety, A an r ample Q-divisor on X, and j=1 Bj an snc divisor on X. I denote by Rr ∼ =V ⊂ DivR X the vector subspace spanned by the components Bj . I write: LV = {B ∈ V | KX + B is log canonical} =

r 1

[0, 1]Bj ;

j=1

EV,A = {B ∈ LV | KX + A + B ∈ Eﬀ X} ⊂ LV . In addition, if S is a component of

%r

j=1

Bj , I write:

S = {B ∈ EV,A | S K⊂ B(KX + A + B)} ⊂ EV,A ; BV,A S=1 = {B = S + B D ∈ EV,A | S K⊂ B(KX + A + S + B D )}. BV,A

Statements. S=1 Theorem 1.12 (Theorem B). BV,A is a rational polytope; moreover: S=1 = {B = S + B D ∈ EV,A | multS F(KX + A + B) = 0}. BV,A

Theorem 1.13 (Theorem C). EV,A is a rational polytope. More precisely, EV,A is the convex hull of ﬁnitely many rational vectors KX +A+Bi where, for all i: Bi ≥ 0 is a Q-divisor, and there is a positive integer pi > 0 such that |pi (KX + A + Bi )| = K ∅. S Theorem 1.14 (Theorem B + ). BV,A is a rational polytope. Furthermore one can say the following. There is an integer r > 0 such that:

• Suppose that B ∈ LV and no component of B is in B(KX +A+B). 7If p(KX + A + B)G is an integral divisor, then no component of B is in Fix rp(KX + A + B) . • Suppose that B ∈ EV,A . If p(KX +A+B) is an integral divisor, then |rp(KX + A + B)| = K ∅.

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The logic of the proof. Here is a table of logical dependencies in Lazi´c’s paper where, e.g., An signiﬁes ‘Theorem A in dimension n:’ An + Cn ⇒ Bn+

(1)

An−1 + Cn−1 ⇒ Bn

(2)

An−1 + Cn−1 + Bn ⇒ Cn

(3)

+ Bn−1

+ Bn ⇒ An

(4)

In this note, I outline all the key steps and ideas in the proof of the last + implication Bn−1 + Bn ⇒ An , stopping somewhat short of a complete proof. The other implications have similar and easier proofs. For a direct analytic proof of a weaker form of Theorem C, see also [P˘au08]. 1.4. The key ideas of the proof. The proof is based on two key ideas that I explain in a bit more detail below and then fully in the rest of the note. The ﬁrst is to prove ﬁnite generation of strictly dlt rings (see below) by restriction to a boundary divisor using lifting lemmas and induction on dimension. The second idea is what I call below the “main construction.” Starting with a klt big adjoint ring with characteristic system D : C → DivR X, I inﬂate the cone C and semigroup Λ to a larger cone C D and semigroup ΛD , and then “chop” into ﬁnitely many smaller Cj and Λj such that the rings R(X; Λj ) are strictly dlt. This is a version of constructions that are ubiquitous in the proofs of all the fundamental theorems of Mori theory. Finite generation of R(X; Λ) follows easily from ﬁnite generation of the R(X; Λj ). Restriction of strictly dlt rings. Deﬁnition 1.15. I say that a dlt big adjoint ring R(X; D) with characteristic system 7 G D(λ) = r(λ) KX + Δ(λ) , where Δ(λ) = A + B(λ) is strictly dlt if there is a prime divisor S that appears in all B(λ) with multiplicity one: B(λ) = S + B D (λ). 7 G I say that R(X; D) is plt if it is strictly dlt and all X, S + B(λ) are plt. When R(X; D) is a strictly dlt adjoint ring, it is natural to want to study the restriction homomorphisms: Q Q 7 G3 7 G3 ρλ : H 0 X; r(λ) KX + A + S + B(λ) → H 0 S; r(λ) KS + Ω(λ) 7 G where Ω(λ) = A + B(λ) |S . The ρλ are not surjective, but, with small additional assumptions, lifting lemmas give us a good control on the images.

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+ Theorem 1.16. Assume Theorems Bn−1 and Bn ; let dim X = n and let R(X; D) be a strictly dlt big adjoint ring on X. The restricted ring:

RS (X; D) =

M

Image(ρλ ) ⊂ R(S; D(λ)|S )

λ∈Λ

is a klt big adjoint ring. Remark 1.17. In fact, RS (X; D) is a klt big adjoint ring not on S, but on some birational model T → S. This technicality is relevant in the proof of the theorem, but it is otherwise unimportant. The techniques of Lazi´c’s proof of Theorem 1.16 are subtle but generally well understood by the experts. I sketch the key ideas in section 5 below. The main construction. In section 3, I give a complete proof that Theorem 1.16 implies Theorem A. Roughly speaking, here is the outline: I want to show that a given dlt adjoint ring R(X; Λ), where Λ = C ∩ Zr ⊂ Rr , satisfying the additional assumption of Theorem A, is ﬁnitely generated. First, I inﬂate C to a larger cone C ⊂ C D ⊂ Rr and extend D : Λ → Div+ Q X to an appropriate DD : ΛD = C D ∩ Zr → Div+ X. Q Next, I construct a decomposition into subcones: C D = ∪rj=1 Cj such that, for all j = 1, . . . , r, writing: D Dj = D|Λ : Λj = Cj ∩ Zr → Div+ Q, j

the ring: G 7 Rj = R(X; Λj ) is strictly dlt with Dj (λ) = rj (λ) Bj + BjD (λ) . Finally, each of the Rj has a surjective restriction homomorphism to a restricted ring: ρj : Rj (X; Λj ) → RBj (X; Λj ), and a relatively straightforward argument shows, assuming—as I may by Theorem 1.16 and induction on dimension—that the restricted rings RBj (X; Λj ) are ﬁnitely generated, that the ring R(X; ΛD ) also is ﬁnitely generated, and then ultimately so is the ring R(X; Λ). The construction is explained in detail in section 3. Acknowledgements. I thank Paolo Cascini, J¨ urgen Hausen, Anne-Sophie Kaloghiros, Vlad Lazi´c, James Mc Kernan and the referee for valuable comments and suggestions.

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2. Natural operations with divisorial rings I brieﬂy discuss the behaviour of divisorial and adjoint rings under natural operations. These properties are elementary and mostly well-known [AH06, ADHL10]. 2.1. Veronese subrings. Deﬁnition 2.1. If R = ⊕λ∈Λ Rλ is a Λ-graded ring (e.g., R could be a divisorial ring), L ⊂ Zr is a ﬁnite index subgroup and ΛD = Λ ∩ L, then I say that . RD = Rλ ⊂ R λ∈Λ"

is a Veronese subring. Remark 2.2. If RD ⊂ R is a Veronese subring, then R is ﬁnitely generated if and only if RD is. Indeed, RD ⊂ R is the ring of invariants under the action of the ﬁnite group G = Zr /L, so the statement is a special case of a well-known theorem of E. Noether. 2.2. Inﬂating. A more general version of the following lemma can be found in [ADHL10, Proposition 1.1.6]. Lemma 2.3. Consider an inclusion of ﬁnite rational cones: C ⊂ C D ⊂ Rr ; and write ΛD = C D ∩ Zr , Λ = C ∩ Zr . Let RD = ⊕λ∈Λ" RλD be a ΛD -graded ring, and write . RλD . R= λ∈Λ

If RD is ﬁnitely generated, then so is R. Proof. This is elementary and well-known, so I only give a very brief sketch of the proof. The cone C ⊂ C D is cut out by ﬁnitely many rational hyperplanes; working one hyperplane at a time, I may assume that C = {w ∈ C D | f (w) ≥ 0},

for a linear map f : Rr → R

with f (Zr ) ⊂ Z. Now the ΛD -grading on RD means that RD has a T = C× r -action, and f : Zr → Z corresponds to a 1-parameter C× → T, in turn endowing RD with a Z-grading, and then, tautologically: . D = RnD . R = R+ n≥0 D Now R+ ⊂ RD is ﬁnitely generated, because it is the subring of invariants for the action of the reductive group C× on . . D z n Rm RD [z] = z n RD = n≥0 D with weight −n + m. acting on z n Rm

n∈Z≥0 , m∈Z

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2.3. Injective characteristic systems. In general, the characteristic system D : Λ → DivQ X of a divisorial ring is not injective. Lemma 2.4. Consider a characteristic system D : Λ → DivQ X where Λ = C ∩ Zr for a ﬁnite rational cone C ⊂ Rr . Assume that D is the restriction of a rational linear function, still denoted D : Rr → DivR X. Write C = D(C) ⊂ DivQ X, the image of C under D, and Λ = C ∩ DivZ X ⊂ DivZ X. Then R(X; Λ) is ﬁnitely generated if and only if R(X; Λ) is ﬁnitely generated. Proof. A simple application of all the above. 2.4. Q-linear equivalence. Deﬁnition 2.5. Let X be a projective normal variety. Denote by Div0R X ⊂ DivR X the subgroup of divisors that are R-linearly equivalent to 0. Two characteristic systems on X: D : Λ → DivQ X and DD : Λ → DivQ X are Q-linearly equivalent if there exists an additive map div : Λ → Div0Q X such that D(λ) = DD (λ) + div(λ) for all λ ∈ Λ. Remark 2.6. If D and DD are Q-linearly equivalent, then R(X; D) and R(X; DD ) have isomorphic Veronese subrings. In particular, one is ﬁnitely generated if and only if the other is. In some arguments, I use this device to replace the ample Q-divisor A by a Qlinearly equivalent Q-divisor AD such that AD is “general,” in the sense that AD ≥ 0, the coeﬃcients of AD are as small as I care for them to be, and AD meets every divisor and locally closed locus in sight as generically as possible. Lemma 2.7. Let X be nonsingular projective and R(X; D) a big adjoint ring on X. • If R(X; D) is dlt, then there exists a Q-linearly equivalent system DD such that R(X; DD ) is a klt big adjoint ring. • If R(X; D) is strictly dlt, then there exists a Q-linearly equivalent system DD such that R(X; DD ) is a plt big adjoint ring. Sketch of Proof. The idea is, of course, to “absorb” into A a small amount of B(λ) where it has coeﬃcient 1. I brieﬂy discuss a very special case that illustrates the key issue. Assume that Λ = N2 = Ne1 + Ne2 and D(e1 ) = KX + A + S1 , D(e2 ) = KX + A + S2 where S1 , S2 are smooth and meet transversally. The ring R(X; D) is dlt.

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For N D 0 we can write: N A ∼ S1 + T 2 ∼ S2 + T 1 where S1 + S2 + T1 + T2 is an snc divisor. Choose a rational function ϕ ∈ k(X) such that −S1 + T1 = −S2 + T2 + divX ϕ. For 0 < ε > 1 we have: D=−

ε ε ε ε ε S1 + T 1 = − S2 + T 2 + divX (ϕ). N N N N N

Then AD = A − D is ample and, setting Bi = (ε/N )Ti : 7 εG K X + A + S 1 = K X + A D + S 1 + D = K X + AD + 1 − S1 + B1 N 7 G ε ε divX (ϕ). K X + A + S 2 = K X + AD + 1 − S2 + B2 + N N Next, deﬁne a new characteristic system DD : N2 → DivQ X by 7 εG DD (e1 ) = KX + AD + 1 − S1 + B1 N G 7 ε DD (e2 ) = KX + AD + 1 − S2 + B2 N By construction, DD is Q-linearly equivalent to D and the ring R(X; DD ) is klt, which proves the ﬁrst part of the statement in this case. 2.5. Proper birational morphisms. Adjoint rings behave well under proper birational morphisms; when working with the restriction of strictly dlt rings, it is useful to blow up X to simplify singularities in order to satisfy the assumptions of the lifting lemma. Lemma 2.8. Let X be nonsingular projective and R(X; D) a plt big adjoint ring on X: 7 G D(λ) = r(λ) KX + Δ(λ) where Δ(λ) = A + S + B(λ). Let B ⊂ X be an snc divisor such that all Supp B(λ) ⊂ B. There is a proper birational morphism f : Y → X, and a plt big adjoint ring R(Y ; DD ): 7 G DD (λ) = r(λ) KY + ΔD (λ) where ΔD (λ) = AD + T + B D (λ) with T ⊂ Y the proper transform of S, with the following properties: • The f -exceptional set E is a divisor. Also, denoting by B D ⊂ Y the proper transform of B ⊂ X, B D ∪ E is an snc divisor and all Supp B D (λ) ⊂ B D ∪ E. • For all λ ∈ Λ, Q 3 KY + T + AD + B D (λ) = f p KX + S + A + B(λ) + E(λ), where E(λ) ≥ 0 is f -exceptional. This implies that R(X; D) = R(Y ; DD ).

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7 G • For all λ ∈ Λ, the pair T, B D (λ)|T is terminal. Proof in a special case. I prove the statement in a special case that contains all the ideas: assume that Λ = N, that is: . 7 G R(X; D) = H 0 X; n(KX + S + A + B) n≥0

where (X, S + B) is a plt pair. For f : Y → X a proper birational morphism with exceptional divisors Ei , I write M KY + T + fp−1 B = f p (KX + S + B) + ai Ei with all ai > −1. Here ai = a(Ei ; KX +S +B) is the discrepancy along Ei of the divisor KX +S +B: it only depends on the geometric valuation % ν = ν(Ei ) measuring order of vanishing along Ei . Next, setting BY = fp−1 B − ai <0 ai Ei , I get: KY + T + BY = f p (KX + S + B) + E % where fp BY = B and E = a≥ 0 ai Ei ≥ 0 is exceptional. Pick a good resolution f : Y → X with the property that all geometric valuations ν with a(ν; KX + S + B) < 0 are divisors on Y (the set of these valuations is ﬁnite hence such a resolution exists); it is a simple matter to check that the pair (T, BY |T ) is terminal. Finally, choose an ample Q-divisor M AD = f p A − ε i Ei on Y , where 0 < εi > 1. Setting B D = BY + is terminal, and:

%

D εi Ei , it is still true that (T, B|T )

KY + T + AD + B D = f p (KX + S + A + B) + E.

3. The main construction In this section I give a complete proof of Theorem A assuming Theorem 1.16. %r Lemma 3.1. Let %rX be a nonsingular projective variety, i=1 Bi an snc divisor on X, and B = i=1 bi Bi a klt divisor (that is, 0 ≤ bi < 1 for i = 1, . . . , r). Let A be an ample Q-divisor on X and assume that for some integer p > 0 |p(KX + A + B)| = K 0. Consider the parallelepiped: B=

r 1 i=1

[bi , 1]Bi ⊂ DivR X,

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Alessio Corti

and the cone and monoid: 7 G C = R+ KX + A + B ⊂ DivR X;

Λ = C ∩ DivZ X.

Then, assuming Theorem 1.16, the ring R(X; Λ) is ﬁnitely generated. Proof. In the course of the proof, I work in the vector subspace V = Rr ⊂ DivR X spanned by the prime divisors Bi ; I denote by D : V n→ DivR X the canonical inclusion. By suitably enlarging (Lemma 2.3) the set {Bi }, and an appropriate choice of the canonical divisor (Lemma 2.7), I can assume that KX + A +

r M i=1

bi Bi =

r M

pi Bi

i=1

where all pi ≥ 0. In addition, perhaps by blowing up%X and using Lemma 2.8, I r can assume that, even after enlargement, the divisor i=1 Bi is still snc. The key is to “chop up” R = R(X; Λ) into ﬁnitely many strictly dlt subrings. Consider the r ‘back faces’ of the parallelepiped B: M Bj = {B = Bj + ci Bi | all bi ≤ ci ≤ 1} for j = 1, . . . , r. i9=j

It is clear that C=

r '

Cj ,

where

7 G Cj = R+ KX + A + Bj

j=1

%r r (this %r uses in a crucial way that C ⊂ i=1 R+ Bi ) hence, setting Λj = Cj ∩ Z , R = j=1 Rj where each Rj = R(X; Λj ) is a strictly dlt adjoint ring. For j = 1, . . . , r, denote by ρj : Rj → RBj = RBj (X; Λj ) the surjective ring homomorphisms to the restricted rings. By Theorem 1.16, and by induction on dimension, the RBj are ﬁnitely generated. I show that R is ﬁnitely generated. I can’t prove this directly, so let σj ∈ H 0 (X; Bj ) be a section vanishing on Bj (σj is determined up to multiplication by a nonzero constant); I show instead that the ring R[σ1 , . . . , σr ],

graded by

Nr ,

is ﬁnitely generated. By Lemma 2.3 again, this implies that R itself is ﬁnitely generated. Fix the total degree function τ : Nr → N,

τ (m1 , . . . , mr ) =

r M

mj .

j=1

Let N D 0 be large enough that the following holds: If m = (m1 , . . . , mr ) ∈ Cj ∩ Zr and τ (m) > N, then m − Bj = (m1 , . . . , mj − 1, . . . , mr ) ∈ C ∩ Zr . (5)

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Finite generation of adjoint rings after Lazi´c: an introduction

(It is pretty obvious that you can ﬁnd N D 0 with this property; draw a picture!) Prepare now the following ﬁnite sets: • a basis G0 of ⊕τ (λ)≤N Rλ ; • for all j = 1, . . . , r, a set Gj ⊂ Rj ⊂ R lifting a set of generators of RBj . I conclude the argument by showing that, using equation 5, the union G = ∪rj=0 Gj ∪ {σ0 , . . . , σr } generates the ring R[σ1 , . . . , σr ]. It is enough to show that R ⊂ C[G]. Assume by induction that M ≥ N and: for τ (λ) ≤ M all Rλ ⊂ C[G]; let τ (λ) = M + 1, and x ∈ Rλ . Now, for some j = 1, . . . , r, λ ∈ Cj , so consider the restriction homomorphism: ρj : R j → R B j . It is clear that there is xj ∈ C[Gj ] such that ρj (x − xj ) = 0. This means that x − xj = σj y 0

where y ∈ H (X; D(λ − Bj )). By property 5, λ − Bj ∈ C, therefore y ∈ Rλ−Bj

has total degree

τ (y) = τ (x) − 1,

hence, by induction, y ∈ C[G], and hence also x ∈ C[G]. Proof of Theorem A. I need to show that a big klt adjoint ring R = R(X; D), where D(C) ⊂ Div+ R X, is ﬁnitely generated. By a simple application of Lemma 2.3, I may assume that D : (C ⊂ Rr ) → Div+ R X is rational linear. By Lemma 2.4, I may also assume that D is injective. As before for v ∈ C I write r 3 Q M 7 G bi (v)Bi . D(v) = r(v) KX + A + B(v) = r(v) KX + A + i=1

Let el for l = 1, . . . , m be generators of the cone C. I will shortly need the quantity δ = min min{1 − bi (v)} = min i=1,...,r v∈C

min {1 − bi (el )} > 0.

i=1,...,r l=1,...,m

For a rational vector v ∈ C lying F on the hyperplane Π = {v | r(v) 7= 1}, consider the parallelepiped B(v) = [bi (v), 1] ⊂ V and the cone C(v) = R+ KX + G A + B(v) . Because all sides of all parallelepipeds B(v) have length ≥ δ, there are ﬁnitely many vectors v1% , . . . , vn ∈ C ∩ Π such that C ⊂ ∪nk=1 C(vk ). Write n r Λ(v) = C(v)∩Z . Now R ⊂ k=1 R(X; Λ(vk )) is ﬁnitely generated by Lemma 2.3 and Lemma 3.1.

4. Lifting lemmas A quick internet search will turn up several papers on lifting lemmas. The prototype can be traced back to [Siu98]; the best place to start learning the material is [Laz04, Theorem 11.5.1]; the lifting theorem 4.2, Theorem 4.5 and corollary 4.7 are all due to Hacon and Mc Kernan.

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4.1. General initial set-up. All variants and improvements of the lifting lemma have a common initial set-up that I now summarise: • X is nonsingular projective; S ⊂ X is a nonsingular divisor; • (X, Δ = S+A+B) is a plt pair; here A is an ample Q-divisor; I always assume that A meets transversally everything in sight, and I sometimes assume that the coeﬃcients of A are suﬃciently small; • Write Ω := (A + B)|S ; I assume that the pair (S, Ω) is terminal. The purpose of the lifting lemma is always this: Fix a strictly positive integer p such that pΔ ∈ DivZ X, then study the restricted adjoint linear system: |p(KX + Δ)|S . Now, of course, p(KX + Δ)|S = p(KS + Ω), and, in general, I don’t expect the restricted linear system to be the complete linear system |p(KS + Ω)|. Indeed, simple examples show that, for basic reasons, the restricted linear system can have a ﬁxed part. The key point of the lifting lemma is to choose divisors Θ, Φ on S with 0 ≤ Θ ≤ Ω and Θ + Φ = Ω and compare the restricted linear system |p(KX +Δ)|S with the linear system with ﬁxed part |p(KS + Θ)| + Φ. Notation 4.1. Let Ei be prime divisors on X. For divisors D1 =

M

di1 Ei ,

I write D1 ∧ D2 =

D2 =

M

M

di2 Ei ,

min{di1 , di2 }Ei .

4.2. Simple lifting. This is the simplest statement that one can make: Theorem 4.2. Fix an integer p > 0 such that pΔ is integral. Assume that S K⊂ B(KX + Δ). Write Fp =

1 Fix |p(KX + Δ)|S ; p

Φp = Ω ∧ Fp ;

Θp = Ω − Φp .

Then |p(KX + Δ)|S = |p(KS + Θp )| + pΦp .

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Finite generation of adjoint rings after Lazi´c: an introduction

4.3. Sharp lifting. Next I state a subtle but crucial improvement of the lifting lemma. Deﬁnition 4.3. Let X be normal projective, S ⊂ X a codimension 1 subvariety, and D a Q-Cartier divisor on X. If S K⊂ B(D), then the restricted asymptotic ﬁxed part is 7 G 1 FS (D) = inf Fix |nD|S ∈ Div+ R S. 0 0 such that pΔ is integral. Assume that S K⊂ B(KX + Δ + A/p). (Note that this holds in particular if S ⊂ K B(KX + Δ + εA) for some rational 0 ≤ ε ≤ 1/p.) Write FS = FS (KX + Δ + A/p). Consider a Q-divisor Φ on S such that pΦ

is integral and

Ω ∧ FS ≤ Φ ≤ Ω;

write

Θ = Ω − Φ.

Then |p(KX + Δ)|S ⊃ |p(KS + Θ)| + pΦ.

Remark 4.6. Sharp lifting improves simple lifting in two ways: it relaxes the assumption and it strengthens the conclusion. It relaxes the assumption. Here I just require that 7 G S⊂ K B KX + Δ + A/p . It strengthens the conclusion. The conclusion now holds for Ω ∧ FS ≤ Φ whereas earlier I required Ω ∧ Fp = Φp : note that FS = FS (KX + Δ + A/p) ≤ FS (KX + Δ) ≤ 7 G 1 ≤ Fix |p(KX + Δ)|S = Fp , p hence Φ is allowed potentially to be smaller than Fp .

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Alessio Corti

4.4. Tinkering lifting. It is possible still to tinker with the statement of the lifting lemma: Corollary 4.7. Fix an integer p > 0 such that pΔ is integral. Assume that S K⊂ B(KX + Δ). Write FS = FS (KX + Δ), and ﬁx a rational ε > 0 such that ε(KX + Δ) + A is ample. Consider a Q-divisor Φ on S such that Q ε3 FS ≤ Φ ≤ Ω; write Θ = Ω − Φ. pΦ is integral and Ω ∧ 1 − p Then

|p(KX + Δ)|S ⊃ |p(KS + Θ)| + pΦ.

Proof. Corollary 4.7 follows from Theorem 4.5: Q 3 ε3 1Q 1 (KX + Δ) + ε(KX + Δ) + A , KX + Δ + A = 1 − p p p hence: Q 1 3 Q ε3 1 FS KX + Δ + A ≤ 1 − FS (KX + Δ) + FS (ample) = p p p Q ε3 FS (KX + Δ); = 1− p thus, the assumptions (those pertaining to Φ) of Theorem 4.5 are satisﬁed, hence its conclusion holds.

5. Restriction of strictly dlt rings In this ﬁnal section, I sketch the proof of Theorem 1.16. Let me brieﬂy recall the set-up. X is nonsingular projective, dim X = n, and R(X; D) is a strictly dlt big adjoint ring on X with characteristic system: 7 G D(λ) = r(λ) KX + Δ(λ) where Δ(λ) = S + A + B(λ). The aim, remember, is to show that the restricted ring RS (X; D) =

. λ∈Λ

Image(ρλ ),

where

Q Q 7 G3 7 G3 ρλ : H 0 X; r(λ) KX + Δ(λ) → H 0 S; r(λ) KS + Ω(λ)

is the restriction map, is a klt adjoint ring. After some simple manipulations, I may in addition assume the following: G 7 (1) All S, B(λ)|S are terminal pairs. (This can be achieved by using Lemmas 2.7 and 2.8 in tandem.)

Finite generation of adjoint rings after Lazi´c: an introduction

213

(2) A has small coeﬃcients and meets everything in sight as generically as possible; in particular, for instance, I assume that all pairs Q 7 G 3 S, Ω(λ) = A + B(λ) |S are terminal. (See Remark 2.6 for this.) 7 G (3) For λ ∈ Λ, S ⊂ K B D(λ) 7. ThisG can be achieved as an application of TheS=1 orem Bn : C D = C ∩ R+ BV,A is a ﬁnite rational cone; now work with D D D Λ = Λ ∩ C and R(X; Λ ) in place of Λ and R(X, Λ): the point is that RS (X; ΛD ) = RS (X; Λ). %r (4) Denote by ei ∈ Rr the standard basis vectors. Then C = i=1 R+ ei ⊂ Rr is a simplicial cone, Λ = Nr ⊂ Rr , and D : Λ → DivQ X is the restriction of a linear function that, abusing notation, I still denote by D : Rr → DivR X. This can be achieved by ﬁnding a triangulation of C on which D is linear. Notation 5.1. Below I denote by Π ⊂ Rr the aﬃne hyperplane spanned by the basis vectors ei . By what I said, Δ, B : Λ → Div+ Q X are restrictions of functions that, abusing notation, I still denote by Δ, B : Rr → DivR X. These are degree 0 homogeneous; hence, they are determined by their restrictions to the aﬃne hyperplane Π ⊂ Rr ; note that these restrictions are aﬃne. Similarly, Ω : Rr → DivR S is degree 0 homogeneous, and Ω|Π is aﬃne. 7 G Lemma 5.2. For λ ∈ Λ, write FS (λ) = FS D(λ) (N.B. by construction if Z _ n > 0, then FS (nλ) = nFS (λ)). Then FS (−) can be uniquely extended to a degree 1 homogeneous convex function that, abusing notation, I still denote by FS : C → Div+ R S. FS is continuous on the interior Int C but not necessarily on C. Proof. By homogeneity I extend to FS : C ∩ Qr → Div+ R S; this function is homogeneous convex hence locally Lipschitz hence locally uniformly continuous hence it can uniquely be extended to a function on C continuous on Int C. After some further blowing up, I may in addition assume: (5) There is a ﬁxed snc divisor F on S such that, for all λ ∈ Λ, Supp FS (λ) ⊂ F . Now for w ∈ C write: 7 G Ω(w) = A + B(w) |S ;

Φ(w) = Ω(w) ∧ FS (w);

Θ(w) = Ω(w) − Φ(w).

By construction, for all w ∈ C, 0 ≤ Θ(w) ≤ Ω(w) and Θ(w) + Φ(w) = Ω(w).

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Alessio Corti

Lemma 5.3. For all λ ∈ Λ, there is Z _ n = n(λ) > 0 such that Φ(λ) = Ω(λ) ∧

7 G 1 Fix |D(nλ)|S . n

In particular, for all λ ∈ Λ, Θ(λ) ∈ Div+ Q S is a rational divisor (and so is Φ(λ)). Proof. The proof is explained very well in [HM09, Theorem 7.1]; it is an application of tinkering lifting; it is simpler than and based on the same idea of the proof of Lemma 5.4 below. The proof of Theorem 1.16 follows easily if I show that Θ|Π∩C : Π ∩ C → Div+ R S 7 G is piecewise aﬃne. (Indeed, write DS (λ) = r(λ) KS + Θ(λ) . By Lemma 5.3 and sharp lifting, the restricted ring RS (X; D) and the adjoint ring R(S, DS ) have a common Veronese subring.) I don’t prove the statement completely. Instead, in the remaining part of this section, I prove: Lemma 5.4. Let x ∈ Π(R) and assume that the smallest rationally deﬁned aﬃne subspace U ⊂ Rr containing x is Π. Then, Θ|Π∩C is aﬃne in a neighbourhood of x. This is compelling, but note that it stops short of proving Theorem 1.16: the statement implies that there is a decomposition of C in rational subcones such that Θ is aﬃne on each subcone, but there is no guarantee that the decomposition is locally ﬁnite, nor indeed that the subcones themselves are ﬁnite. The proof of the lemma contains all the ideas of Lazi´c’s proof of Theorem 1.16. Lemma 5.5 (Diophantine approximation). Let x ∈ Rn ; denote by U the smallest rationally deﬁned aﬃne subspace containing x, and let dim U = m − 1. Fix ε > 0 and an integer M > 0. There exist vectors w1 , . . . , wm ∈ U (Q) with the following properties: (1) For i = 1, . . . , m, there are real numbers 0 < ri < 1

with

m M

ri = 1,

i=1

x=

m M

ri wi ;

i=1

(2) there is an m-tuple (p1 , . . . , pm ) of strictly positive integers, all pi divisible by M , such that all pi wi ∈ Zn are integral, and ||x − wi || < ε/pi .

Proof of Lemma 5.4. There is a nagging diﬃculty with the proof:

Finite generation of adjoint rings after Lazi´c: an introduction

215

A nagging diﬃculty and an additional assumption. The point is this: by deﬁnition, Φ(−) = Ω(−) ∧ FS (−) and Θ(−) = Ω(−) − Φ(−); + hence, although Ω : C → Div+ R S is linear, and FS : C → DivR S is convex, Φ(−) is not necessarily convex, and Θ(−) is not necessarily concave. To be more speciﬁc, if for some prime divisor P ⊂ S, multP Ω(x) = multP FS (x), then Θ(−) may fail to be concave in a neighbourhood of x. I ﬁrst run the proof under the following additional assumption:

For all prime divisors P ⊂ S,

multP Ω(x) K= multP FS (x).

In the proof below, I only use the additional assumption to ensure that Θ(−) is concave in a neighbourhood of x. At the end, I brieﬂy explain how to get around this diﬃculty. The strategy of the proof. The idea of the proof is to choose a real ε > 0, an integer M > 0 and a rational Diophantine approximation 7 G (wi , Θi ) ∈ Π(Q) × DivQ (S) of the vector x, Θ(x) ∈ Π(R) × DivR (S), such that i. For i = 1, . . . , r, there are real numbers 0 < µi < 1 with

6 %r x = i=1 µi wi , µi = 1, and %r Θ(x) = i=1 µi Θi . i=1

r M

% µi Δ(wi ). (All this is guaranteed In particular this implies that Δ(x) = by Lemma 5.5.) In addition I require that: ii. Θi ≤ Θ(wi ). Indeed, once I know this, then, by concavity of Θ: M M µi Θ(wi ) ≤ Θ(x) = µi Θi . % I deduce Θ(x) = µi Θ(wi ) and, by concavity again, this implies that Θ is aﬃne on the convex span of the wi . Choice of ε. I choose ε > 0 small enough that it has the following features: (a) Θ is concave in {w | ||w−x|| < ε}. (This is OK by the additional assumption.) (b) There is a local Lipschitz constant C = Cx such that: If ||w − x|| < ε,

then ||Θ(w) − Θ(x)|| < C||w − x||.

(All concave functions are locally Lipschitz.)

216

Alessio Corti

(c) There is a constant 0 < δ < 1 with the following property: For all prime divisors P ⊂ S: 7 G If multP Ω(x) − Θ(x) > 0, and ||w − x|| < ε,

7 G then multP Ω(w) − Θ(w) > δ,

and I also assume that ε < δ. (d) If p ≥ 1 and ||w − x|| < pε , then the Q-divisor Δ(w) − Δ(x) +

A p

is ample. (e) If ||w − x|| < ε, then the Q-divisor 3 εQ (C + 1) KX + Δ(w) + A δ is ample. (f) If Θ ∈ DivQ S and ||Θ − Θ(x)|| < ε, then no component of Θ is in the asymptotic ﬁxed part F(KS + Θ). + that I can arrange for this to It is a simple consequence of Theorem Bn−1 hold.

Choice of M > 0. Next I choose M > 0 such that the rational Diophantine approximation given by Lemma 5.5: 7 G (wi , Θi ) ∈ Π(Q) × DivQ (S) of the vector x, Θ(x) ∈ Π(R) × DivR (S), satisﬁes the following conditions: (1) For i = 1, . . . , r, there are real numbers 0 < µi < 1 with

6 %r x = i=1 µi wi , µi = 1, and %r Θ(x) = i=1 µi Θi . i=1

r M

In particular this implies that Δ(x) =

%

µi Δ(wi ).

(2) There is an r-tuple (p1 , . . . , pr ) ∈ Nr of positive integers such that: • (pi wi ; pi Θi ) ∈ Nr × DivZ (S) is integral; • for all i, ||x − wi || <

ε pi

and ||Θ(x) − Θi || <

ε . pi

Finite generation of adjoint rings after Lazi´c: an introduction

217

(3) For all prime divisors P ⊂ S: • If multP Θ(x) < multP Ω(x), then also multP Θi < multP Ω(wi ) (this is automatic from feature (c)); • If multP Θ(x) = multP Ω(x), then also multP Θi = multP Ω(wi ). • If multP Θ(x) = 0, then also multP Θi = 0. (Although the second bullet point doesn’t strictly speaking follow from a blind usage of Lemma % 5.5, it is easy to arrange for it to hold. Indeed in this case multP Θ(x) = µi multP Ω(wi ) and it pays to declare from the start that multP Θi = multP Ω(wi ). The third bullet point is similar and easier.) (4) M is large enough that the pi are large enough and divisible enough that: F(KS + Θi ) = Fix |pi (KS + Θi )| . + .) (This can easily be arranged using Theorem Bn−1

The key inclusion. For all i = 1, . . . , r I show the key inclusion: ! Q ! Q 3! Q 3 3! ! ! ! ! !pi KX + Δ(wi ) ! ⊃ !pi KS + Θi ! + pi Ω(wi ) − Θi . S

(6)

The key inclusion allows me to control the restricted algebra in a neighbourhood of x: as I show below, it readily implies that Θi ≤ Θ(wi ). If you get bored with the details of the proof, you may want to press forward to the conclusion. I plan to prove this using sharp lifting. To begin with, I remark that I am in the general initial set-up of section 4.1. For all i, I now check that the speciﬁc assumptions of sharp lifting are satisﬁed; that is: Ω(wi ) ∧ Fi ≤ Ω(wi ) − Θi 7 G where Fi = FS KX + Δ(wi ) + A/pi . For all prime divisors P ⊂ S I check that 7 G 7 G multP Ω(wi ) ∧ Fi ≤ multP Ω(wi ) − Θi . The discussion breaks down in two cases: Case 1: multP Θ(x) = multP Ω(x). By condition 3, one has that multP Θi = multP Ω(wi ). By feature (d), Δ(wi ) − Δ(x) + A/pi is ample, therefore: Q A3 = multP FS KX + Δ(wi ) + pi Q Q A 33 ≤ = multP FS KX + Δ(x) + Δ(wi ) − Δ(x) + pi 7 G ≤ multP FS KX + Δ(x) = 0.

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Alessio Corti

Case 2: multP Θ(x) < multP Ω(x). Using feature (e): Q A3 ≤ multP Fi = multP FS KX + Δ(wi ) + pi Q 7 G (C + 1) ε 3 ≤ 1− multP FS KX + Δ(wi ) = pi δ Q (C + 1) ε 3 = 1− multP FS (wi ). pi δ This implies that Q 7 G (C + 1) ε 3 multP Ω(wi ) − Θ(wi ) multP Ω(wi ) ∧ Fi ≤ 1 − pi δ (indeed, by deﬁnition Ω(wi ) − Θ(wi ) = Ω(wi ) ∧ FS (wi )); and, ﬁnally, using feature (c) and: ||Θ(wi ) − Θi || ≤ ||Θ(wi ) − Θ(x)|| + ||Θ(x) − Θi || ≤ C

ε ε ε + = (C + 1) , pi pi pi

in tandem, I get: Q

1−

7 G (C + 1) ε 3 multP Ω(wi ) − Θ(wi ) ≤ pi δ 7 G C +1 ≤ multP Ω(wi ) − Θ(wi ) − ε≤ p 7 7 i G G ≤ multP Ω(wi ) − Θ(wi ) + multP Θ(wi ) − Θi = 7 G = multP Ω(wi ) − Θi .

Conclusion. I show that the key inclusion of equation 6 implies the statement. By construction of the function Θ : C → Div+ R S and the lifting lemma, I know that: ! 7 ! 7 G! G! !pi KX + Δ(wi ) ! = !pi KS + Θ(wi ) ! + Φ(wi ). S

Thus, the key inclusion readily implies that: ! 7 ! 7 G! G! Mob !pi KS + Θi ! ≤ Mob !pi KS + Θ(wi ) ! .

(7)

Now, by feature (f), no component of Θi is in the ﬁxed part of |pi (KX + Θi )|; thus, the last equation implies Θi ≤ Θ(wi ).

Finite generation of adjoint rings after Lazi´c: an introduction

219

How to remove the additional assumption. Assume that for some prime P ⊂ S multP Ω(x) = multP FS (x). Consider an eﬀective divisor G=

r M

εj Bj .

j=1

If the coeﬃcients 0 ≤ εj are small enough, then:

7 G • Writing B D (λ) = B(λ) + G > B(λ), all the S, B D (λ)|S are terminal, and: • A − G is still ample, so I can choose AD ∼Q A − G meeting everything in sight transversally, and such that, upon setting 7 G ΔD (λ) = S + AD + B D (λ); ΩD (λ) = AD + B D (λ) |S , 7 G then all the pairs S, ΩD (λ) are terminal. Note that, from the deﬁnition, for all λ ∈ Λ: 7 G 7 G FS (λ) = FS D(λ) = FS DD (λ) .

By choosing AD generically, I can arrange that the additional assumption for DD is satisﬁed, and conclude as above that ΘD (−) is rational aﬃne in a neighbourhood of x. By construction: 7 G in a neighbourhood of x, multP ΩD (−) − ΘD (−) = multP FS (−), that is, in a neighbourhood of x, multP FS (−) also is rational aﬃne. But then x ∈ U = {w | multP Ω(w) = multP FS (w)} implies that, in a neighbourhood of x, multP Ω(w) = multP FS (w) (U is aﬃne and deﬁned over Q, hence Π ⊂ U by minimality of Π), that is, Θ(−) is concave in a neighbourhood of x after all.

References [ADHL10] Ivan Arzhantsev, Ulrich Derenthal, Juergen Hausen, and Antonio Laface. Cox rings, arXiv:1003.4229. [AH06]

Klaus Altmann and J¨ urgen Hausen. Polyhedral divisors and algebraic torus actions. Math. Ann., 334(3):557–607, 2006.

[BCHM09] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., posted on November 13, 2009. PII: S 0894-0347(09)00649-3 (to appear in print). [CL10]

Alessio Corti and Vladimir Lazi´c. Finite generation implies the Minimal Model Program, arXiv:1005.0614.

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Alessio Corti, editor. Flips for 3-folds and 4-folds, volume 35 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2007.

[ELM+ 06] Lawrence Ein, Robert Lazarsfeld, Mircea Mustat¸˘ a, Michael Nakamaye, and Mihnea Popa. Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble), 56(6):1701–1734, 2006. [FA92]

Flips and abundance for algebraic threefolds. Soci´et´e Math´ematique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Ast´erisque No. 211 (1992).

[FM00]

Osamu Fujino and Shigefumi Mori. A canonical bundle formula. J. Diﬀerential Geom., 56(1):167–188, 2000.

[HM06]

Christopher D. Hacon and James McKernan. Boundedness of pluricanonical maps of varieties of general type. Invent. Math., 166(1):1–25, 2006.

[HM09]

Christopher D. Hacon and James McKernan. Existence of minimal models for varieties of log general type II. J. Amer. Math. Soc., posted on November 13, 2009. PII: S 0894-0347(09)00651-1 (to appear in print).

[Laz04]

Robert Lazarsfeld. Positivity in algebraic geometry. II, volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals.

[Laz09]

Vladimir Lazi´c. Adjoint rings are ﬁnitely generated, arXiv:0905.2707.

[Nak04]

Noboru Nakayama. Zariski-decomposition and abundance, volume 14 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2004.

[P˘ au08]

Mihai P˘ aun. Relative critical exponents, non-vanishing and metrics with minimal singularities, arXiv:0807.3109.

[Sho03]

Vyacheslav V. Shokurov. Prelimiting ﬂips. Tr. Mat. Inst. Steklova, 240(Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry):82–219, 2003.

[Siu98]

Yum-Tong Siu. Invariance of plurigenera. Invent. Math., 134(3):661–673, 1998.

Alessio Corti, Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Log canonical thresholds on varieties with bounded singularities Tommaso de Fernex∗, Lawrence Ein†, Mircea Mustat¸a˘‡

Abstract. We consider pairs (X, A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a ﬁxed set that satisﬁes the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a ﬁxed aﬃne space deﬁned by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisﬁes the Ascending Chain Condition. 2010 Mathematics Subject Classiﬁcation. Primary 14E15; Secondary 14B05, 14E30. Keywords. Log canonical threshold, ascending chain condition.

1. Introduction The log canonical threshold is a fundamental invariant in birational geometry. It is attached to a divisor with real coeﬃcients on a variety with mild singularities. An outstanding conjecture due to Shokurov [18] predicts that in any ﬁxed dimension, if the coeﬃcients of the divisors are taken in any given set of positive real numbers satisfying the descending chain condition (DCC), then the set of all possible log canonical thresholds satisﬁes the ascending chain condition (ACC).1 This conjecture has attracted considerable attention due to its implications to the Termination of Flips Conjecture. More precisely, Birkar showed in [1] the following: if Shokurov’s conjecture is known in dimension n, and if the log Minimal Model Program is known in dimension (n − 1), then there are no inﬁnite sequences of ﬂips in dimension n for pairs of non-negative log Kodaira dimension. We note that due to the results in [2], Termination of Flips is the remaining piece in order to establish the log Minimal Model Program in arbitrary dimension. There is another outstanding open problem in the area, the Abundance Conjecture, but the circle of ideas we are discussing does not have anything to say in that direction. Shokurov’s conjecture was proved in the case of smooth (and, more generally, locally complete intersection) ambient varieties in [3], building on work from [4] and [10]. In this note we deal with the more general case of varieties that have bounded singularities, in a sense to be explained below. ∗ Partially

supported by NSF CAREER grant DMS-0847059. supported by NSF grant DMS-0700774. ‡ Partially supported by NSF grant DMS-0758454, and by a Packard Fellowship. 1 A set of real numbers satisﬁes DCC (respectively, ACC) if it does not contain any inﬁnite sequence that is strictly decreasing (respectively, strictly increasing). For short, such a set will be called a DCC set (respectively, ACC set). † Partially

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Let k be an algebraically closed ﬁeld of characteristic zero. We assume that our ambient varieties are deﬁned over k, and are normal and Q-Gorenstein. Let X be any such variety. Instead of dealing with R-divisors, we work in the more general setting of R-ideals on X: these are formal products A = aq11 · · · aqrr , where the ai are nonzero ideal sheaves and the qi are positive real numbers. If the qi lie in a subset Γ of R>0 , we say that A is a Γ-ideal. Given two R-ideals A and B on X and a point x ∈ Supp(A), if (X, B) is log canonical, then one deﬁnes the mixed log canonical threshold lct(X,B),x (A) to be the largest c such that the pair (X, Ac B) is log canonical at x. One reduces to the more familiar setting of log canonical thresholds when B = OX . In order to study limits of log canonical thresholds, the basic ingredient in the methods used in [4, 10, 3] is the construction of generic limit ideals. The main obstruction in proving Shokurov’s Conjecture in its general form comes from the problem of constructing a “generic limit ambient space” where the generic limit ideal should live. From this point of view, the advantage in the smooth and locally complete intersection cases is that one can easily reduce to work with one ﬁxed polynomial ring, so that in the end, in order to construct generic limit ideals, it suﬃces to take a ﬁeld extension and complete the ring at the origin. In this paper we consider the case of bounded singularities. We say that a collection of germs of algebraic varieties (Xi , xi ) has (formally) bounded singularities if there are integers m and N such that for every i there is a subscheme Yi in AN whose ideal is deﬁned by equations of degree ≤ m, and a point yi ∈ Yi such that 3 3 O Xi ,xi @ OYi ,yi . Equivalently, this means that there exists a morphism π : Y → T , such that for every i there is a closed point ti ∈ T and a point yi in the ﬁber Yti 3 3 over ti such that O Xi ,xi @ OYti ,yi . At the moment this appears to be the most general context where the approach through generic limits can be put to work, and it seems likely that new methods will be needed to attack the conjecture in its general form. We can now state our main result. Theorem 1.1. If Γ ⊂ R>0 is a DCC set, then there is no inﬁnite strictly increasing sequence of log canonical thresholds lct(X1 ,B1 ),x1 (A1 ) < lct(X2 ,B2 ),x2 (A2 ) < . . . , where the (Xi , xi ) form a collection of klt varieties with bounded singularities and Ai , Bi are Γ-ideals such that all pairs (Xi , Bi ) are log canonical. The result in [3] covers the case when the Xi are assumed to be nonsingular (or more generally, locally complete intersection), and Γ = Z>0 . In the nonsingular setting, the ﬁrst result in this direction was obtained in [4], where it was shown using ultraﬁlter constructions that every limit of invariants of the form lct(Xi , ai ), with dim(Xi ) = n for all i, is again an invariant of the same form. Koll´ ar replaced in [10] the ultraﬁlter approach by a generic limit construction, using more traditional algebro-geometric methods. In addition, using the results in [2] he proved a semicontinuity property for log canonical thresholds that allowed him to treat a special case of the conjecture, namely when the log canonical threshold of the limit

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is computed by a divisor with center at one point. In [3] we gave a more elementary proof of Koll´ar’s semicontinuity result, and showed that this can be used in fact to deduce the full statement of the above theorem when all Xi are nonsingular (and Γ = Z>0 ). More general cases, such as when the Xi are locally complete intersection or have quotient singularities, were deduced from the nonsingular case in a direct fashion. Regarding the statement of Theorem 1.1, we emphasize that while the category of varieties with bounded singularities is quite large, it is not large enough for the applications to the Minimal Model Program. More precisely, it is not the case that, for example, terminal Q-factorial singularities in a ﬁxed dimension have bounded singularities. This is simply because one can construct quotient singularities that satisfy these properties, and of arbitrary embedding dimension. Furthermore, as Miles Reid pointed out to us, starting from dimension ﬁve there are families of terminal singularities of arbitrary high embedding dimension that are not nontrivial quotients by ﬁnite group actions. The proof of Theorem 1.1 is based on the generic limit construction from [10], suitably adapted to our setting. The main novelty in the proof of the above theorem is the simultaneous construction of a generic limit of the ambient spaces and of the ideal sheaves involved. The fact that the embedded dimension of the varieties is bounded is necessary in order to have the “generic limit variety” being deﬁned by an ideal in a power series ring with ﬁnitely many variables. We use the fact that the singularities themselves are bounded to guarantee that such limit variety is normal, Q-Gorenstein, and klt. There are however several technical diﬃculties that arise when working in this general setting. Some of these technical points are of a more general nature, not necessarily related to the main topic of the paper, and therefore their treatment will be deferred to the end of the paper. This will result in two appendices. The ﬁrst technical diﬃculty comes from the fact that we work with singular varieties in the formal setting. It has became evident since [4] that the formal setting is very natural when dealing with this kind of problems. However, while in the previous papers [4, 10, 3] the formal setting always occurred at regular points, in the present paper we need to work with possibly singular schemes of ﬁnite type over complete local rings. The generic limit construction that is essential for proving Theorem 1.1 requires us to develop the theory of log canonical pairs in a slightly more general framework than usual: working with R-ideals on schemes of ﬁnite type over a complete Noetherian ring (of characteristic zero). This is the case since starting with a sequence as in the theorem, the generic limit construction provides an ambient space that is the spectrum of a complete local ring. While this ring is the completion at a closed point of a scheme of ﬁnite type over a ﬁeld, we need to consider ideals in this ring that do not come via completion from the ﬁnite type level. This will require us to extend the basic results on log canonical thresholds to this setting. In particular, in order to have the notion of relative canonical class in this setting, we will need to develop a theory of sheaves of diﬀerentials that is adapted to this context. This part is extracted from the main body of the paper and forms

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the ﬁrst appendix. The second appendix is devoted to another technical complication arising in the proof of Theorem 1.1. The problem comes from the fact that we need to be able to bound the Gorenstein index of the varieties appearing in the statement in order to conclude that the limit variety is Q-Gorenstein. To this end, we prove a general result on the behavior of the Gorenstein index in bounded families (see Theorem B.1). This result is of independent interest, and a slightly simpliﬁed version of it can be stated as follows. Theorem 1.2. Let f : X → T be a morphism of normal complex varieties such that every ﬁber of f is a normal variety with rational singularities. Then there is a nonempty Zariski open subset T ◦ ⊆ T and a positive integer s such that for every point x ∈ X with t = f (x) ∈ T ◦ , the ﬁber Xt is Q-Gorenstein at x if and only if the total space X is Q-Gorenstein at x; furthermore, in this case both sKX and sKXt are Cartier at x. A variant of the result holding over arbitrary algebraically closed ﬁelds of characteristic zero is given in Theorem B.8. This will be the version of the result applied in the proof of Theorem 1.1. Acknowledgment. We are grateful to Mark de Cataldo for comments and suggestions related to the material in Appendix B, and to J´anos Koll´ ar and Miles Reid for sharing with us some interesting examples of singularities.

2. Log canonical pairs on schemes of ﬁnite type over a complete local ring Throughout this section, let k be a ﬁeld of characteristic zero, and let R = k[[x1 , . . . , xn ]]. Our goal is to deﬁne and prove the basic properties of log canonical and log terminal pairs when the ambient space is a scheme of ﬁnite type over R. Of course, the deﬁnitions are parallel to the ones in the case of schemes of ﬁnite type over ﬁelds. The main diﬀerence is that in order to deﬁne the relative canonical class, we need to work with sheaves of special diﬀerentials as deﬁned in Appendix A. The theory of special diﬀerentials enables us to deﬁne the notion of relative canonical divisor in this setting (see in particular Lemma A.11). Once we have the notion of relative canonical class, the theory of singularities of pairs can be built in the same way as in the case of schemes of ﬁnite type over a ﬁeld, for which we refer to [11]. However, we will need to work with R-ideals (as opposed to R-divisors), hence we give all deﬁnitions in this setting. In the following, letFX be a scheme of ﬁnite type over R. An R-ideal on X is r a formal product A = i=1 aipi , where r is a positive integer, each ai is a nonzero (coherent) ideal sheaf on X, and the pi are positive real numbers. We call A a proper R-ideal if there is an i with ai K= OX . If the pi are required to lie in some subset Γ ⊆ R>0 , then A is called a Γ-ideal. The notions we are interested in are invariant with respect to the equivalence relation that identiﬁes two R-ideals if they have the same order of vanishing along

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all divisorial valuations. More precisely, we consider all proper birational morphisms π : Y → X, with Y normal, and all prime divisors E on Y . Every such E deﬁnes a valuation ordE of the function ﬁeld of X. The image of E on X is the center of E on X, and it is denoted by cX (E). If a is an ideal sheaf on X, then ordE (a) is the minimum of ordE (w), where Fr w varies over the sections of a deﬁned at the generic point of cX (E). If A = i=1 api i is an R-ideal on X, then ordE (A) :=

r M

pi · ordE (ai ).

i=1

The equivalence relation identiﬁes A and AD whenever ordE (A) = ordE (AD ) for every E as above. Remark 2.1. By Theorem 2.3 below, whenever we consider a valuation ordE as above, we may assume that the model Y on which E lies is nonsingular. Fr Example 2.2. If A = i=1 api i , where all pi ∈ Q, then A is identiﬁed F with b1/m , r i . where m is a positive integer such that mpi ∈ Z for all i, and b = i=1 amp i 1/m 1/m Furthermore, two such Q-ideals b1 and b2 are identiﬁed if and only if for some positive integer q, the ideals bq1 and bq2 have the same integral closure. The product of R-ideals the Fr is deﬁned in the obvious way, by concatenating Fr i factors. Similarly, if A = i=1 api i is as above, and q ∈ R>0 , then Aq := i=1 aqp i . Note that these operations preserve the above equivalence classes. Fr Suppose now that X is normal, and let A = i=1 api i beFan R-ideal on X. A r log resolution of (X, A) is a log resolution for the pair (X, i=1 ai ). Recall that this is a proper birational morphism π : Y → X, with Y nonsingular, such that the exceptional locus of π and the inverse images of the subschemes V (ai ) are Cartier divisors, and all these divisors have simple normal crossings. Since we are in characteristic zero, the existence of log resolutions in our setting is guaranteed by the results in [19]. For completeness, we explain how to get log resolutions from the results in loc. cit. The two theorems below hold for arbitrary quasi-excellent schemes2 , so they hold in particular in our setting, for schemes of ﬁnite type over R. Theorem 2.3. ([19]) For every integral scheme X of ﬁnite type over R, there is a proper birational morphism π : Y → X with Y nonsingular. Furthermore, we may construct π such that it is an isomorphism over Xreg . Theorem 2.4. ([19]) If X is a nonsingular scheme as above, and D is an eﬀective divisor on X, then there is a proper birational morphism π : Y → X such that Y is nonsingular and π ∗ (D) has simple normal crossings. Furthermore, we may assume that π is an isomorphism over X 1 Supp(D). Let us explain how to combine these two theorems Fr in order to get log resolutions. Suppose that (X, A) is a pair as above, with A = i=1 api i , and X normal. We ﬁrst 2 A scheme is quasi-excellent if it is covered by aﬃne open subsets of the form Spec(A ), with i each Ai a quasi-excellent ring; the deﬁnition of quasi-excellent ring is similar to that of excellent ring, but one does not require the ring to be universally catenary, see [15, p. 260].

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apply Theorem 2.3 to construct π1 : Y1 → X proper and birational, and such that Y1 is nonsingular. Since X is normal, there is an open subset U ⊆ X such that π1 is an isomorphism over U , and Z := X 1 U has codim(Z, X) ≥ 2. We note that if ϕ : Y → Y1 is proper and birational, with Y nonsingular, and if ϕ−1 (π1−1 (Z)) is a divisor, then Exc(π1 ◦ ϕ) = Exc(ϕ) ∪ ϕ−1 (π1−1 (Z)). (1) In particular, this exceptional locus is a divisor (recall that Exc(ϕ) is a divisor; this follows for instance from Lemma FrA.11). We blow up successively along i=1 ai , and along the inverse image of Z, to get π2 : Y2 → Y1 . We now apply one more time Theorem 2.3 to get a proper and birational morphism π3 : Y3 → Y2 with Y3 nonsingular. Furthermore, we do this so that π3 is an isomorphism over (Y2 )reg . It follows from (1) that E := Exc(π1 ◦ π2 ◦ π3 ) is an eﬀective divisor on Y3 , and we have eﬀective divisors Ei on Y3 such that ai · OY3 = OY3 (−Ei ). Furthermore, if Z D = (π1 ◦ π2 ◦ π3 )−1 (Z), then by construction Supp(Z D ) ⊆ Supp(E) ⊆ Supp(Z D ) ∪ Supp(E1 + · · · + Er ).

(2)

We apply Theorem 2.4 to get a proper birational morphism π4 : Y → Y3 with Y nonsingular, and such that π4∗ (Z D + E1 + · · · + Er ) has simple normal crossings. Furthermore, we may and will assume that this is an isomorphism over Y3 1 Supp(Z D + E1 + · · · + Er ). We let π : Y → X be the composition. Using (1) and (2) we see that Exc(π) is a divisor, and that it is contained in the support of π4∗ (Z D + E1 + · · · + Er ). Therefore π is a log resolution of (X, A). A similar argument can be used to show that any two log resolutions of (X, A) are dominated by a third one. Suppose now that X is Q-Gorenstein, and let π : Y → X be a log resolution Fr of a pair (X, A), where A = i=1 api i . Let KY /X be the relative canonical divisor as deﬁned in Appendix A (cf. Lemma A.11). Since KY /X is supported on the %\ exceptional locus, it follows that there is a simple normal crossings divisor j=1 Ej on Y with KY /X =

\ M

κj Ej ,

a i · OY = OY

j=1

Q

−

\ M

αi,j Ej

3

for 1 ≤ i ≤ r.

(3)

j=1

The pair (X, A) is called log canonical if κj + 1 ≥

r M

αi,j pi = ordEj (A)

(4)

i=1

for all j. If all inequalities in (4) are strict, the pair is Kawamata log terminal (or klt, for short). If A = OX , we simply say that X is log canonical or klt, respectively. Note that the deﬁnitions are independent of the representative for A in our equivalence class. The fact that the deﬁnition is independent of the log

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resolution follows in the same way as in the case of schemes of ﬁnite type over a ﬁeld. The key ingredients are given by Lemma A.11 iii), and the fact that any two log resolutions can be dominated by a third one. Remark 2.5. It follows from Remark A.12 that if X is nonsingular, then the log canonicity of a pair (X, a) is independent of the R-scheme structure of X. Again, it is not clear to us whether the same remains true if X is singular (however, see Remark 2.12 below for one case when this holds, which is the one that concerns us most). Fs Fr Let A = i=1 api i and B = i=1 bqi i be R-ideals on X, with A a proper ideal. If the pair (X, B) is log canonical, then we deﬁne the mixed log canonical threshold lct(X,B) (A) (written also as lctB (A) when there is no ambiguity about the ambient scheme) as the largest c ≥ 0 such that (X, Ac B) is log canonical. If B = OX , then we simply write lct(X, A) or lct(A), and we call it the log canonical threshold of A. % If π as above is a log resolution of (X, A · B), and if bi · OY = OY (− j βi,j Ej ) for 1 ≤ i ≤ s, then %s κj + 1 − i=1 βi,j qi %r lct(X,B) (A) = min . (5) j i=1 αi,j pi Note that since A is assumed to be proper, there are i and j such that αi,j > 0, hence the above minimum is ﬁnite. If Ej is such that the minimum in (5) is achieved, we say that E computes lct(X,B) (A). Fr We also consider a local version of the above invariant. If A = i=1 api i is an R-ideal on X, we denote by Supp(A) the union of the closed subsets of X deﬁned by the ideals ai . If x ∈ Supp(A), and (X, B) is log canonical in some open neighborhood of x, then lct(X,B),x (A) is the largest c ≥ 0 such that (X, Ac B) is log canonical in some neighborhood of x. If B = OX , we write lctx (X, A) or lctx (A). Of course, lct(X,B),x (A) can be described by a formula analogous to (5), in which the minimum is over those j such that x ∈ cX (Ej ). For simplicity, we will state most of the basic properties of log canonical thresholds only in the unmixed setting, since we will only need these versions. The following lemma is a simple consequence of the formula for the log canonical threshold in terms of a log resolution. Fr Lemma 2.6. Suppose that X is log canonical, and let A = i=1 api i and B = Fs qi i=1 bi be proper R-ideals on X. If s ≤ r, and ai ⊆ bi and pi ≥ qi for all i ≤ s, then lct(A) ≤ lct(B). A similar assertion holds for the local version of log canonical thresholds. It is sometimes convenient to reduce the study of log canonical thresholds of Rideals to that of Q-ideals (hence to that of usual ideals). This can be done using the following two lemmas (the ﬁrst one deals with the log canonical threshold, while the second one treats the divisors computing the log canonical threshold). Fr Lemma 2.7. Assume that X is log canonical, and let A = i=1 api i be a proper Rideal on X. If (pi,m )m≥1 are sequences of positive real numbers with limm→∞ pi,m =

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Fr p pi for every i ≤ r, and if Am = i=1 ai i,m , then limm→∞ lct(Am ) = lct(A). A similar assertion holds for the local version of log canonical threshold lctx (A). Proof. The assertion follows immediately from formula (5). Lemma 2.8. Suppose that X is log F canonical, and let E be a divisor computing r lctx (A) = λ, for some R-ideal A = i=1 api i on X, containing x in its support. Then one can ﬁnd sequences of rational (pi,m )m≥1 with limm→∞ pi,m = Fr numbers p λpi , and such that if we put Am = i=1 ai i,m , then lctx (Am ) = 1 and E computes lctx (Am ) for every m. Proof. Let π : Y → X be a log resolution of A such that E is a divisor on Y . With the notation in (3), after restricting to a suitable open neighborhood of x, we may assume that x ∈ cX (Ej ) for all j. Consider the rational polyhedron r M ( R αi,j ui for all j . P = (u1 , . . . , ur ) | κj + 1 ≥ i=1

If E = Ej0 , % then we see that (λp1 , . . . , λpr ) lies on the face PE of P deﬁned by κj0 + 1 = i αi,j0 ui . Since PE is itself a rational polyhedron, it follows that (λp1 , . . . , λpr ) can be written as the limit of a sequence (p1,m , . . . , pr,m ) ∈ PE ∩ Qr . It is clear that this sequence satisﬁes our requirements. The following lemma allows one to reduce the study of the log canonical threshold to the case when this invariant is computed by a divisor with center equal to a closed point. The proof is the same as that of [3, Lemma 5.2], so we omit it. Lemma 2.9. Suppose that X is log canonical, A is a proper R-ideal on X, and x ∈ X is a closed point deﬁned by the ideal mx . If c = lctx (A), then there is a nonnegative real number t such that c = lctx (mtx · A), and this log canonical threshold is computed by a divisor E over X having center equal to x. We will mainly be interested in the case when the ambient variety is either a scheme of ﬁnite type over a ﬁeld, or the spectrum of the completion of the local ring of such a scheme at a closed point. The following proposition gives the compatibility of the log canonical threshold with respect to taking such a completion. Suppose that X is a scheme of ﬁnite type over k and x ∈ X is a closed Fr pi 4 point, and consider g : Z = Spec(O X,x ) → X. If A = i=1 ai is an R-ideal on X, F r p i # the R-ideal we denote by A i=1 (ai OZ ) . We consider the Cartesian diagram W

h

!Y π

f

# 4 Z = Spec(O X,x )

(6)

g

# !X

where π : Y → X is a proper birational morphism with Y nonsingular.

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Remark 2.10. Since g is a regular morphism (see [15, Chapter 32]), it follows that so is h. Recall that a morphism of Noetherian schemes is regular if it is ﬂat, and has geometrically regular ﬁbers. Since g is regular, we see that X is normal around x if and only if Z is normal, and since h is regular, we see that W is nonsingular. Proposition 2.11. With the above notation, the following hold: i) The pair (X, B) is log canonical (respectively, klt) in a neighborhood of x if # is log canonical (respectively, klt). and only if (Z, B) # is a ii) If (X, B) is log canonical in a neighborhood of x ∈ Supp(A), then A proper R-ideal and # lct(X,B),x (A) = lct(Z,B) b (A). iii) Under the assumptions in ii), if Ei and F are as in Remark A.15, then F # computes lct(X,B),x (A) if and only if Ei computes lct(Z,B) b (A). Proof. Note that if π is a log resolution of (X, A · B), then f is a log resolution of # · B), # because h is a regular morphism. Furthermore, if F is a nonsingular (Z, A prime divisor on Y such that x ∈ cX (F ), and E is a component of h∗ (F ), then E is a nonsingular prime divisor on W . It is clear that the coeﬃcient of F in a simple normal crossings divisor D on Y is equal to the coeﬃcient of E in h∗ (D). We now deduce the assertions in the proposition from Proposition A.14. Remark 2.12. In the setting of the proposition, it follows from Proposition A.14 4 that the divisor KW/Z does not depend on the presentation of O X,x via Cohen’s Structure Theorem. Furthermore, if AD is an arbitrary R-ideal on Z (not necessarily coming from X), and if we consider a log resolution W D → Z of (Z, AD ), then it follows from Lemma A.13 and Remark A.12 that KW " /Z is independent of the 4 presentation of O X,x . Therefore, the (mixed) log canonical thresholds on Z are independent of this presentation. Remark 2.13. Note that in the setting of the proposition, if X is nonsingular, then the conclusion of the proposition also holds if the localization is at a nonclosed point. Indeed, when we deal with nonsingular schemes, then we do not need to consider O(KX ), as the divisors KY /X and KW/Z can be computed using the 0th Fitting ideals of the corresponding sheaves of relative diﬀerentials, and h∗ ΩY /X @ ΩW/Z . The following lemma concerns the behavior of singularities of pairs for schemes of ﬁnite type over a ﬁeld under the extension of the ground ﬁeld. In particular, it allows us to reduce the study of singularities of such pairs to the case when the ground ﬁeld is algebraically closed. Lemma 2.14. Let X be a normal scheme of ﬁnite type over a ﬁeld k. If K is a ﬁeld extension of k, and if ϕ : X = X ×Spec(k) Spec(K) → X is the projection, then Fr for every R-ideal A = i=1 aipi on X, and every x ∈ X and x = ϕ(x), we have:

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i) rKX is Cartier at x if and only if rKX is Cartier at x. i) The pair (X, A) is log canonical (respectively, klt) in some neighborhood of x if and only if (X, A) is log canonical (respectively, klt) in some neighborhood of x, where r 1 (ai OX )pi . A= i=1

iii) If X is log canonical at x, then lctx (X, A) = lctx (X, A). iv) If F is a divisor that computes lctx (X, A), and if E is a component of the divisor F = F ×Spec(k) Spec(K) on X whose center contains x, then E computes lctx (X, A). Proof. Note that the ﬁbers of ϕ are disjoint unions of zero-dimensional, reduced schemes. Since we are in characteristic zero and ϕ is ﬂat, we deduce from this fact that ϕ is regular. In particular, X is normal (though it might be disconnected), and ϕ−1 (Xreg ) = X reg . It is also easy to deduce that if F is a reﬂexive sheaf on X, then ϕ∗ (F) is reﬂexive, and F is generated by one element around x if and only if ϕ∗ (F) is generated by one element around x. Since ΩX/k @ ϕ∗ (ΩX/k ), this implies that we can take ϕ∗ (KX ) = KX , and rKX is Cartier at x if and only if rKX is Cartier at x. Suppose now that X is Q-Gorenstein, and let f : Y → X be a log resolution of (X, A). If Y = Y ×Spec(k) Spec(K), then we have a Cartesian diagram Y

ψ

g

# X

!Y f

ϕ

# ! X.

If f is an isomorphism over U ⊆ X, then g is an isomorphism over the dense open subset ϕ−1 (U ) of X. Since ψ is regular, arguing as in the proofs of Propositions A.14 and 2.11, we see that g is a log resolution of (X, A), and we have KY /X = ψ ∗ (KY /X ). Note also that if E is a prime nonsingular divisor on Y such that x ∈ cX (E), then there is a component F of E = ψ ∗ (E) such that x ∈ cX (F ). For every such F , the valuation ordF restricts to the valuation ordE on the function ﬁeld of X. The remaining assertions in the proposition are easy consequences of these observations. We now give some further properties of log canonical thresholds that will be used in the proof of our main result. These generalize corresponding results for schemes of ﬁnite type over a ﬁeld, and for usual ideals. Let us ﬁx the notation. In what follows X is a log canonical scheme of ﬁnite type over a ﬁeld k. Let x ∈ X 4 be a closed point with dim(OX,x ) = n, and let g : Z = Spec(O X,x ) → X be the canonical morphism. We have seen in Proposition 2.11 that if A is an R-ideal on X

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# = lctx (X, A). The following lemma allows such that x ∈ Supp(A), then lct(Z, A) us to approximate every log canonical threshold on Z by log canonical thresholds of pull-backs of R-ideals on X. Fr Proposition 2.15. Let B = i=1 bqi i be an R-ideal Fr on Z. If m is the ideal deﬁning the closed point on Z, and if we put Bd = i=1 (bi + md )qi , then lim lct(Z, Bd ) = lct(Z, B).

d→∞

Furthermore, if there is a divisor E over Z computing lct(Z, B) and with center equal to the closed point, then lct(Z, Bd ) = lct(Z, B) for d D 0. Proof. The proof follows verbatim the proof of [4, Proposition 2.5] (the hypothesis in loc. cit. that the ambient scheme is nonsingular does not play any role). The key step is to show that lct(Z, B) = inf F

ordF (KW/Z ) + 1 , ordF (B)

where the inﬁmum is over the divisors F over Z (lying on some W ) having center equal to the closed point. We refer to loc. cit. for details. Remark 2.16. With the notation in the proposition, if n denotes the maximal d 4 4 d ideal in OX,x , then m = n · O X,x , and OX,x /n @ OX,x /m for every d. It follows that, after possibly replacing X by an aﬃne open neighborhood of x, every R-ideal Id for some R-ideal Ad on X. Bd in the proposition can be written as A Lemma 2.17. Suppose that X is klt. If n is the ideal deﬁning x ∈ X, then lct(X, n) ≤ n. Proof. We apply Lemma 2.14, with K = k, the algebraic closure of k. We see that lct(X, n) = lctx (X, n) for any point x ∈ ϕ−1 (x). Since in some neighborhood of x, n is equal to the ideal deﬁning x, we see that we may replace X by X. Therefore we may assume that k is algebraically closed. As pointed out by Kawakita, the bound now follows from the proof of [9, Theorem 2.2]. For the sake of the reader, we brieﬂy recall the argument. After replacing X by its index one cover corresponding to O(KX ), we may assume that KX is Cartier. Let f : Y % → X be a log resolution of (X, n), and write n · OY = OY (−E), with E = i mi Ei . We can choose F = Ei0 such that O(−E)|F is big and nef. For every m ≥ 1, we have an exact sequence 0 → O(KY /X − mE) → O(KY /X − mE + F ) → O(KF ) ⊗ O(−mE)|F → 0. By the Kawamata-Viehweg Vanishing Theorem, it follows that P (m) := h0 (F, O(KF ) ⊗ O(−mE)|F ) = χ(F, O(KF ) ⊗ O(−mE)|F ), and this is a polynomial of degree (n − 1) since ((−E)n−1 · F ) > 0. It follows that there is an integer s, with 1 ≤ s ≤ n such that P (s) K= 0.

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On the other hand, another application of the Kawamata-Viehweg Vanishing Theorem implies that R1 f∗ O(KY /X − sE) = 0, hence the sequence 0 → f∗ O(KY /X − sE) → f∗ O(KY /X − sE + F ) → f∗ (O(KF ) ⊗ O(−sE)|F ) → 0 is exact. It follows that f∗ O(KY /X − sE) K= f∗ O(KY /X − sE + F ), hence there is a rational function ϕ on X such that D := divY (ϕ) + KY /X − sE + F ≥ 0, and the coeﬃcient of F in D is zero. Since KY /X − sE + F is f -exceptional, it follows that ϕ ∈ O(X). Therefore ordF (ϕ) ≥ 0, and we conclude that ordF (KY /X ) + 1 ≤ s · ordF (n), and hence lct(X, n) ≤ s ≤ n. Lemma 2.18. If a and b are ideals on X such that their supports contain x ∈ X, then lctx (a + b) ≤ lctx (a) + lctx (b). Proof. Arguing as in the proof of the previous lemma, we may assume that k is algebraically closed. It is now convenient to use the language of multiplier ideals, for which we refer to [14, Chapter 9]. The version of the Summation Theorem from [7, Corollary 2] implies that for every λ ≥ 0 we have the following description for the multiplier ideals of exponent λ of a sum of ideals: M J (X, (a + b)λ ) = J (X, aα bβ ). (7) α+β=λ

Recall that lctx (a) is the smallest α such that x lies in the support of J (X, aα ). Let c1 = lctx (a) and c2 = lctx (b). It is enough to show that x lies in the support of J (X, (a+b)c1 +c2 ). This follows from (7), since given α, β such that α+β = c1 +c2 , then either α ≥ c1 , or β ≥ c2 . In the ﬁrst case we have J (X, aα bβ ) ⊆ J (X, aα ) ⊆ J (X, ac1 ), whose support contains x. The case β ≥ c2 is similar.

Fr Proposition 2.19. Suppose that X is klt in a neighborhood of x. Let B = i=1 bqi i be a proper R-ideal on Z, and let m be the ideal deﬁning the closed point of Z. n i) If bi ⊆ msi for every i, then lct(Z, B) ≤ s1 q1 +···+s . r qr Fr ii) Suppose that A = i=1 aqi i is another R-ideal on Z, and let ε > 0 be a real number. If d is a positive integer such that d ≥ εqni and ai + md = bi + md for all i, then | lct(A) − lct(B)| ≤ ε.

Proof. For i), it is clear that if s = s1 q1 + · · · + sr qr , then lct(Z, B) ≤ lct(Z, ms ) =

lct(Z, m) . s

By Proposition 2.11 we have lct(Z, m) = lct(X, n), where n is the ideal deﬁning x ∈ X, and we conclude by Lemma 2.17.

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For ii), we ﬁrst show that if a and b are proper ideals on Z such that a+m\ = b+ m , then | lct(a)−lct(b)| ≤ n/]. Note that by Proposition 2.15, it is enough to show that | lct(a+mN )−lct(b+mN )| ≤ n/] for all N ≥ ]. Therefore we may assume that there are ideals aD and bD on X such that a = a#D and b = b#D . By Proposition 2.11, it is enough to show that if aD + n\ = bD + n\ , then | lctx (aD ) − lctx (bD )| ≤ n/]. We deduce from Lemmas 2.17 and 2.18 that \

lctx (aD ) ≤ lctx (aD + n\ ) = lctx (bD + n\ ) ≤ lctx (bD ) +

n lct(n) ≤ lctx (bD ) + . ] ]

The inequality lctx (bD ) ≤ lctx (aD ) + n\ follows by symmetry. We now prove ii). After writing each qi as a decreasing limit of rational numbers, we see using Lemma 2.7 that it is enough to prove ii) when all qi ∈ Q. Let us choose a positive integer p such that all pqi are integers. It is enough to show that lct(B) ≥ lct(A) − ε, as the other inequality will follow by symmetry. After replacing each ai by ai + md , we may assume that ai = bi + md . Therefore aipqi =F(bi + md )pqi , and thisFideal has the same integral i i + mdpqi . Since i (bipqi + mdpqi ) and i bpq have the same image closure as bpq i i \ mod m , where ] = dp · mini qi , we deduce G 7F pqi 7F pqi G + mdpqi ) = p · lct lct(A) = p · lct i (bi i ai 7 7F pqi G n G ≤ lct(B) + ε, ≤ p · lct + i bi ] where the last inequality follows from the assumption that ] ≥ np/ε. The following result is a key ingredient in the proof of our main result. In the case of schemes of ﬁnite type over a ﬁeld and usual ideals, it was proved in [10] and [3]. Fr Fr Proposition 2.20. Consider the proper R-ideals A = i=1 aiqi and B = i=1 bqi i on Z, and suppose that E is a divisor that computes lct(A), having center equal 4 to the closed point of Z. If ai + pi = bi + pi for all i, where pi = {u ∈ O X,x | ordE (u) > ordE (ai )}, then lct(A) = lct(B). We will need the following lemma. 4 Lemma 2.21. If E is a divisor over Z = Spec(O X,x ) with center equal to the closed point of Z, then the restriction of ordE to the function ﬁeld of X is of the form ordF for some divisor F over X with center x. Moreover, one can ﬁnd a Cartesian diagram as in (6) such that F appears as a prime nonsingular divisor on Y and E appears as h∗ (F ). Proof. We ﬁrst note that the valuation ring Ov of v = ordE is essentially of ﬁnite 4 type over O X,x , and its residue ﬁeld k(E) has transcendence degree (n − 1) over 4 k (recall that n = dim(OX,x ) = dim(O X,x )). Indeed, let us realize E as a prime divisor on some nonsingular T , with ϕ : T → Z birational. By assumption, E lies in the ﬁber T0 of T over the closed point in Z. Note that T0 is a scheme of

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ﬁnite type over K, where K = k(x), hence over k (recall that K is ﬁnite over k). Let y ∈ E be a closed point. It follows from the Dimension Formula (see [15, Theorem 15.6]) that dim(OT,y ) = n. Since E is a prime divisor on T , we deduce that Ov = OT,E is essentially of ﬁnite type over k. Its residue ﬁeld k(E) is the fraction ﬁeld of OE,y . This has dimension (n − 1), hence k(E) has transcendence degree (n − 1) over K (equivalently, over k). We now show that we can ﬁnd a sequence of schemes Z0 , . . . , Zm , with Z0 = Z and each Zi being the blow-up of Zi−1 at the center Ci−1 of ordE on Zi−1 , such that the center of ordE on Zm has codimension one. This is well-known in the case of schemes of ﬁnite type over a ﬁeld, and for example the proof of [12, Lemma 2.45] can be easily adapted to our setting. Indeed, arguing:as in loc. cit. one ﬁrst shows that if the Zi are constructed as above, then Ov = i≥1 OZi ,Ci . If y1 , . . . , yn−1 ∈ Ov are such that their residues give a transcendence basis of k(E) over K, let i be large enough such that all the yj lie in OZi ,Ci . Therefore the residue ﬁeld of OZi ,Ci has transcendence degree (n − 1) over K. Another use of the Dimension Formula 4 implies that dim(OZi ,Ci ) = 1. Since O X,x is a Nagata ring, so is OZi ,Ci , hence the normalization S of OZi ,Ci is ﬁnite over OZi ,Ci . Now S is a Dedekind ring, and Ov is the localization of S at a maximal ideal. If m is such that OZm ,Cm contains S, then we see that codim(Cm , Zm ) = 1. We similarly construct a sequence of varieties Xi , where X0 = X and Xi is the X blow-up of Xi−1 along the center Ci−1 of the restriction w of ordE to the function ﬁeld of X. It follows by induction on i that we have Cartesian diagrams Zi # Zi−1

gi

gi−1

! Xi # ! Xi−1

such that Ci = (gi )−1 (CiX ) (this follows since CiX is clearly the closure of gi (Ci ), and since Ci lies over the closed point in Z). Since gm induces an isomorphism X X between Cm and Cm , it follows that codim(Cm , Xm ) = 1. Therefore w is a divisorial valuation. The last assertion in the lemma follows by taking a model Y over X X on which the proper transform of Cm is nonsingular. Proof of Proposition 2.20. If k is algebraically closed, then the assertion for usual ideals on Spec(OX,x ) follows from [3, Theorem 1.4] (see also [10]). The assertion extends to the case when k is not algebraically closed via Lemma 2.14. We now extend this to the case of (usual) ideals A and B on Z (that is, we assume r = 1 and q1 = 1). By Lemma 2.21 above, there is a divisor F over X, with center x, such that ordF is equal to the restriction of ordE to the function ﬁeld of X. Furthermore, it follows from the lemma and Proposition 2.11 that given an ideal AD on X with x ∈ Supp(AD ), F computes lctx (AD ) if and only if E computes lct(A#D ). Recall that n is the ideal deﬁning x ∈ X, and m is the ideal deﬁning the closed point in Z. For every d ≥ 1, after replacing X by any aﬃne neighborhood of x, we

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235

ID = B + md . Since ID = A + md and B can ﬁnd ideals ADd and BDd on X such that A d d d ordE (m) ≥ 1, it follows that ordE (A + m ) = ordE (A) for d > ordE (A). Let us ﬁx such a d. Since E computes lct(A), we deduce that lct(A + md ) ≤ lct(A), and if equality holds, then E computes lct(A + md ). On the other hand, the inclusion A ⊆ A + md implies lct(A) ≤ lct(A + md ). We conclude that lct(A) = lct(A + md ), and E computes lct(A + md ). Therefore F computes lct(ADd ) = lct(A). Since in a neighborhood of x, the ideals BDd and ADd are congruent modulo {u | ordF (u) > ordF (ADd )}, we deduce from the case we already know that lct(B+md ) = lct(BDd ) = lct(A). This holds for all d > ordE (A), hence it follows from Proposition 2.15 that lct(B) = lct(A). Let us deduce now the statement of the proposition in the case of R-ideals. We ﬁrst note that the hypothesis implies that ordE (ai ) = ordE (bi ) for all i. Claim. For all positive integers ]1 , . . . , ]r , we have r 1 i=1

b\i i + J\1 ,...,\r =

r 1 i=1

ai\i + J\1 ,...,\r ,

(8)

\1 \r 4 where J\1 ,...,\r = {f ∈ O X,x | ordE (f ) > ordE (a1 · · · ar )}. After replacing each ai and bi by ]i copies of itself, we see that it is enough to prove the claim when all ]i = 1. If ui ∈ bi , let us write ui = wi + hi , with wi ∈ ai , and hi ∈ pi . In this case,   r r r 1 1 1 1 M h m · ui − (9) wi = uj  . wi · i=1

i=1

m=1

i<m

j>m

% Since each of the terms on the right-hand side of (9) has order > m ordE (am ), we deduce the inclusion “⊆” in (8), and the opposite inclusion follows by symmetry. This proves the claim. We now apply Lemma 2.8 to get sequences of positive rational numbers (qm,j )j for 1 ≤Fj ≤ r, with limm→∞ qm,j = lct(A) · qj and, moreover, such that, for all q r m, lct( i=1 ai m,i ) = 1 and this log canonical threshold is computed by E. We choose for each m a positive integer Nm such that Nm qm,j ∈ Z for all j. If we Fr Fr N q N q put a(m) := i=1 ai m m,i and b(m) := i=1 bi m m,i , then lct(a(m) ) = 1/Nm is computed by E. The above claim, together with the case we already know (for usual ideals) gives lct(b(m) ) = 1/Nm for every m. We now deduce lct(B) = lct(A) from Lemma 2.7, which completes the proof of the proposition.

3. Generic limit constructions In this section we give a variant of the generic limit construction from [10] (see also [3]), that allows us to deal with the fact that in Theorem 1.1 the ambient space is not ﬁxed. Let us ﬁx an algebraically closed ﬁeld k of characteristic zero. Let R = k[[x1 , . . . , xN ]], let m be the maximal ideal in R, and for every ﬁeld extension K/k let RK = K[[x1 , . . . , xN ]] and mK = mRK .

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For every d ≥ 1, we consider the quotient homomorphism R → R/md . We identify the ideals in R/md with the ideals in R containing md . Let Hd be the Hilbert scheme parametrizing the ideals in R/md , with the reduced scheme structure. Since dimk (R/md ) < ∞, Hd is an algebraic variety. For every ﬁeld extension K/k, the K-valued points of Hd correspond to ideals in RK /mdK . Mapping an ideal in R/md to its image in R/md−1 gives a surjective map τd : Hd → Hd−1 . This is not a morphism. However, by generic ﬂatness we can cover Hd by ﬁnitely many disjoint locally closed subsets such that the restriction of τd to each of these subsets is a morphism. Let us ﬁx a positive integer m. We also consider a parameter space G for ideals in k[x1 , . . . , xN ] generated by polynomials of degree ≤ m, and vanishing at the origin (we consider on G the reduced structure). Each such ideal is determined by its intersection with the vector space of polynomials of degree ≤ m, hence G is an algebraic variety. Given a ﬁeld extension K/k, the K-valued points of G are in bijection with the ideals in K[x1 , . . . , xN ] vanishing at the origin and generated in degree ≤ m. We now ﬁx also a positive integer r. Consider the product Zd := G × (Hd )r and the map td : Zd → Zd−1 that is given by the identity on the ﬁrst component, and by τd on the other components. As in the case of τd , even though td is not a morphism, we can cover Zd by disjoint locally closed subsets such that the restriction of td to each of these subsets is a morphism. In particular, for every irreducible closed subset Z ⊆ Zd , the map td induces a rational map Z ''( Zd−1 . We now describe the generic limit construction. The main diﬀerence with the treatment in [3] is coming from the ﬁrst factor in Zd . Suppose that we have (r + 1) (r) (1) sequences (pi )i∈I0 , (ai )i∈I0 , . . . , (ai )i∈I0 indexed by I0 = Z+ . Each pi is an ideal in k[x1 , . . . , xN ] generated in degree ≤ m and vanishing at the origin, and each (j) ai is an ideal in k[[x1 , . . . , xN ]]. We consider sequences of irreducible closed subsets Zd ⊆ Zd for d ≥ 1 with the following properties: (v) For every d ≥ 1, the projection td+1 induces a dominant rational map ϕd+1 : Zd+1 ''( Zd . (1)

(r)

(vv) For every d ≥ 1, there are inﬁnitely many i with (pi , ai +md , . . . , ai +md ) ∈ Zd , and the set of such (r + 1)-tuples is dense in Zd . Given such a sequence (Zd )d≥1 , we deﬁne inductively nonempty open subsets Zd◦ ⊆ Zd , and a nested sequence of inﬁnite subsets I0 ⊇ I1 ⊇ I2 ⊇ · · · , (1)

(r)

as follows. We put Z1◦ = Z1 and I1 = {i ∈ I0 | (pi , ai + m, . . . , ai + m) ∈ Z1◦ }. (1) ◦ For d ≥ 2, let Zd◦ = ϕ−1 + d (Zd−1 ) ⊆ Domain(ϕd ) and Id = {i ∈ I0 | (pi , ai (r)

md , . . . ai + md ) ∈ Zd◦ }. It follows by induction on d that Zd◦ is open in Zd , and condition (vv) implies that each Id is inﬁnite. Furthermore, it is clear that Id ⊇ Id+1 .

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Sequences (Zd )d≥1 satisfying (v) and (vv) can be constructed as in [3, Section 4]. Suppose now that we have a sequence (Zd )d≥1 with these two properties. The rational maps ϕd induce a nested sequence of function ﬁelds k(Zd ). Let : K := d≥1 k(Zd ). Each morphism Spec(K) → Zd ⊆ Zd corresponds to an (r + 1)(1) (r) tuple () pd , ) ad , . . . , ) ad ). All ) pd are equal, and we denote this ideal in K[x1 , . . . , xN ] by p. This is generated in degree ≤ m and vanishes at the origin. Furthermore, the compatibility between the morphisms Spec(K) → Zd implies that there are (j) (unique) ideals a(j) in RK (for 1 ≤ j ≤ r) such that ) ad = a(j) + mdK for every (1) (r) d. We call the (r + 1)-tuple (p, a , . . . , a ) the generic limit of the given (r + 1) sequences of ideals. We record in the next lemma some easy properties of this construction. The proof is straightforward, so we omit it. Lemma 3.1. With the above notation, for every j with 1 ≤ j ≤ r, the following hold. (j)

= b for some ideal b ⊆ R and every i, then a(j) = bRK .

(j)

⊆ m for every i, then a(j) ⊆ mK .

i) If ai

ii) If ai

iii) If a(j1 ) , . . . , a(js ) = (0), then for every q ≥ 1 there are inﬁnitely many d such (j ) that ad α ⊆ mq for 1 ≤ α ≤ s. Let Id be the universal ideal on Hd × AN k , whose restriction to the ﬁber over a point in Hd corresponding to the ideal b containing md is equal to b. Similarly, let J be the universal ideal on G × AN k . Let βj be the composition of the embedding N N Zd × AN n→ Z × A with the projection Zd × AN d k k k → Hd × Ak if j K= 0 (or N N Zd × Ak → G × Ak when j = 0) induced by the projection Hdr → Hd onto the j th factor (respectively by the projection G × Hdr → G). We consider the (0) (j) −1 ideals on Zd × AN (J ) and Id = (βj )−1 (Id ) for k deﬁned as follows: Id = (β0 ) (j) 1 ≤ j ≤ r. For a not necessarily closed point z ∈ Zd , we denote by (Id )z the (j) (1) (r) restriction of Id to the ﬁber over z. Each tuple (pi , ai + md , . . . , ai + md ), with (j) (j) i ∈ Id , corresponds to a closed point td,i ∈ Zd such that (Id )td,i · R = ai + md for (0) 1 ≤ j ≤ r, and (Id )td,i = pi . By construction, the set Td := {td,i | i ∈ Id } is dense (j) in Zd . Similarly, if ηd ∈ Zd is the generic point, we have (Id )ηd · RK = a(j) + mdK (0) for j ≥ 1, and (Id )ηd · K[x1 , . . . , xN ] = p. We denote by σd : Zd → Zd × AN k the morphism given by σd (u) = (u, 0). (j)

Lemma 3.2. With the above notation, suppose that pi R ⊆ ai for every i ∈ I0 and every j with 1 ≤ j ≤ r. For every d ≥ 1 there is a nonempty open subset Ud (0) (j) of Zd such that Id ⊆ Id over Ud for 1 ≤ j ≤ r. In particular, pRK ⊆ a(j) . (j)

Proof. Since the support of Id lies in σd (Zd ), it follows that the open subset of (0) (j) Zd × AN k where Id is contained in Id is the inverse image of an open subset Ud in Zd . This is nonempty, since by assumption all td,i , with i ∈ Td , lie in Ud . Since

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ηd ∈ Ud , we deduce that p ⊆ (a(j) + md ) ∩ K[x1 , . . . , xN ]. This holds for every d, hence p · RK ⊆ a(j) . (1)

(r)

Suppose now that the sequences (pi )i∈I0 , (ai )i∈I0 , . . . , (ai )i∈I0 satisfy the hypothesis in Lemma 3.2. In this case we put Wi = Spec(R/pi R) and W = (j) (j) Spec(RK /pRK ). We denote by ai = ai /pi R the ideal on Wi corresponding to (j) ai , for 1 ≤ j ≤ r. It follows from Lemma 3.2 that we may consider the ideals a(j) = a(j) /pRK on W , for 1 ≤ j ≤ r. We denote by mi and mK the ideals deﬁning the closed points in Wi and, respectively, in W . The following is the main technical result about generic limits in our setting. (j)

Proposition 3.3. With the above notation, if all Wi are klt, and all ai ideals, then the following hold.

are proper

i) W is klt. ii) For every d there is an inﬁnite subset Id◦ ⊆ Id such that for all nonnegative real numbers p1 , . . . , pr , and for every i ∈ Id◦ 7 Fr 7 Gpj G 7 Gpj G Fr 7 (j) lct W, j=1 a(j) + mdK = lct Wi , j=1 ai + mdi . Fr iii) If E is a divisor over W computing lct(W, j=1 (a(j) )pj ) for some nonnegative real numbers p1 , . . . , pr , and having center equal to the closed point, then there is an integer dE such that for every d ≥ dE the following holds: there is an inﬁnite subset IdE ⊆ Id◦ , and for every i ∈ IdE a divisor Ei over Wi Fr (j) computing lct(Wi , j=1 (ai + mdi )pj ), such that ordE (mK ) = ordEi (mi ) and (j)

ordE (a(j) + mdK ) = ordEi (ai + mdi ) for every j. In particular, Ei has center the closed point of Wi .

We emphasize that in part iii) of the proposition, both dE and the sets IdE depend on p1 , . . . , pr , and Ei depends on d. Proof. With the notation in Lemma 3.2, let Xd be the closed subscheme of Ud ×AN k (0) deﬁned by Id , and let f : Xd → Ud be the morphism induced by projection. It (j) follows from Lemma 3.2 that we can deﬁne ideal sheaves bd on Xd as the quotient (j) (0) of (the restrictions to Xd of) Id by Id . We denote by (Xd )ξ the ﬁber of Xd (j) (j) over a (not necessarily closed) point ξ ∈ Ud , and by (bd )ξ the restriction of bd (j) to (Xd )ξ . We will apply Corollary B.9 to f and to the ideals bd , with 1 ≤ j ≤ r. Note that each Wi is obtained by completing at σd (td,i ) the ﬁber (Xd )td,i . Since Wi is klt, it follows from Proposition 2.11 that (Xd )td,i is klt at σd (td,i ). Corollary B.9 implies that (Xd )ηd is klt at σd (ηd ). Therefore the base extension to K of this generic ﬁber is klt at the origin (see Lemma 2.14), hence its completion W is klt by another application of Proposition 2.11. This proves i).

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The assertion in ii) follows directly from Proposition 2.11 and Corollary B.9 (see also Remark B.10). In order to prove iii), note ﬁrst that if dE > ordE (a(j) ) for all j with 1 ≤ j ≤ r, then E also computes 7 F G 7 F G lct W, j (a(j) + mdK )pj = lct W, j (a(j) )pj for all d ≥ dE (the inequality “≤” follows since the assumption on d implies ordE (a(j) + mdK ) = ordE (a(j) ) for all j, while the reverse inequality follows from the inclusions a(j) ⊆ a(j) + mdK ). By Lemma 2.21, the restriction of the valuation ordE to the fraction ﬁeld of K[x1 , . . . , xN ] is equal to ordF , for some divisor F over AN K . We can choose dE large enough, so that for d ≥ dE the divisor F comes by base extension from a divisor Fd deﬁned over k(Zd ). It follows from F (j) p Proposition 2.11 and Lemma 2.14 that Fd computes lctσd (ηd ) ((Xd )ηd , ( j bd )ηdj ). The assertion in iii) now follows from Corollary B.9 ii). This completes the proof of the proposition. Corollary 3.4. With the notation and assumption in Proposition 3.3, for every sequence (id )d≥1 with id ∈ Id◦ , and for all nonnegative real numbers p1 , . . . , pr we have 7 Fr G 7 G Fr (j) lct W, j=1 (a(j) )pj = lim lct Wid , j=1 (aid )pj . d→∞

G 7 Fr (j) In particular, if the sequence lct(Wi , j=1 (ai )pj i≥1 is convergent, then it conFr verges to lct(W, j=1 (a(j) )pj ). Proof. Since W and all Wi are klt, it follows from Proposition 2.19 ii) that given N for every j such that pj > 0, then we have any ε > 0, if d ≥ εp j ! 7 F (j) p G G! 7 F j ! lct W, − lct W, j (a(j) + mdK )pj ! ≤ ε, ja ) ! 7 G! 7 F (j) pj G F (j) ! lct Wi , − lct Wid , j (aid + mdid )pj ! ≤ ε. d j (aid ) It follows from the choice of Id◦ in Proposition 3.3 that ! 7 F (j) p G 7 F (j) G! j ! lct W, − lct Wid , j (aid )pj ! ≤ 2ε. j (a ) This gives the assertion in the corollary. Remark 3.5. The argument in the proof of the above corollary can also be carried out if some a(j) is zero; in this case, one has to apply partQi) in Proposition 2.19, in7 F (j) G3 stead of part ii). In particular, we see that if the sequence lct Wi , j (ai )pj converges to a positive real number, then all a(j) , with 1 ≤ j ≤ r, are nonzero.

i≥1

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4. Proof of the main result Our goal is to give a proof of Theorem 1.1. Let us ﬁx an algebraically closed ground ﬁeld k of characteristic zero. Consider a collection (Xi , xi ) of schemes of ﬁnite type over k, with xi ∈ Xi closed points. We say that the family has bounded singularities if there are positive integers m and N such that for every i there is a closed subscheme Yi of AN k whose ideal is deﬁned by polynomials of degree ≤ m, 3 3 and a closed point yi ∈ Yi such that O Xi ,xi @ OYi ,yi . Remark 4.1. The above condition is equivalent with the existence of a morphism π : Y → T of schemes of ﬁnite type over k such that for every i there is a closed 3 3 point ti ∈ T and a closed point yi in the ﬁber Yti over ti such that O Xi ,xi @ OYti ,yi . Indeed, if the collection of varieties has bounded singularities, then it is enough to take T to be a parameter space parametrizing closed subschemes of AN k deﬁned by ideals generated in degree ≤ m, and let Y n→ AN × T be the universal subscheme. k : : Conversely, given π we can ﬁnd ﬁnite aﬃne open covers T = j Vj and Y = j Uj such that π(Uj ) ⊆ Vj for every j. It is enough to take N and m such that each Uj can be embedded as a closed subscheme of AN k , with the ideal generated in degree ≤ m. We now set the notation for the rest of this section. Let us ﬁx N and m, and 4 let XN,m be the set of all klt schemes of the form Spec(O X,x ), where X is a closed subscheme of AN deﬁned by an ideal generated by polynomials of degree ≤ m and k x ∈ X is a closed point. After a suitable translation we may always assume that x is the origin in AN k . We will freely use the notation introduced in the previous section in the construction of generic limits. The following is our main result. Theorem 4.2. With the above notation, if Γ ⊂ R+ is a DCC subset, then there is no inﬁnite strictly increasing sequence of log canonical thresholds lct(W1 ,B1 ) (A1 ) < lct(W2 ,B2 ) (A2 ) < · · · , where all Wi ∈ XN,m , and Ai , Bi are Γ-ideals on Wi , with (Wi , Bi ) log canonical. Proof. Let us assume that there is a strictly increasing sequence as in the stateN 3 ment, with Wi = Spec(O Xi ,0 ) klt, where Xi is a closed subscheme of Ak deﬁned by an ideal pi ⊂ k[x1 , . . . , xN ] generated in degree ≤ m. Let ci = lct(Wi ,Bi ) (Ai ). F si F ri (j) (j) (bi )qi,j where all pi,j , qi,j ∈ Let us write Ai = j=1 (ai )pi,j , and Bi = j=1 (j)

(j)

Γ. We may and will assume that all ai and bi vanish at 0 (though it may (j) (j) happen that Bi = OWi , in which case si = 0). Let ai and bi be the ideals in (j)

(j)

(j)

(j)

R = k[[x1 , . . . , xn ]] such that ai = ai /pi R and bi = bi /pi R. Since Γ satisﬁes DCC, it follows that there is an ε > 0 such that pi,j , qi,j > ε for every i and j. Since Wi is klt around 0, it follows from Proposition 2.19 that N dim(Wi ) ≤ ci ≤ lct(Wi , Ai ) ≤ %ri . εri j=1 pi,j

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This clearly implies that the sequence (ci )i≥1 is bounded, and therefore it converges to some c ∈ R>0 . It also implies that the sequence (ri )i≥1 is bounded. Therefore, after passing to a subsequence we may assume that ri = r for all i. Similarly, since lct(Wi , Bi ) ≥ 1 for every i, it follows that we may assume that si = s for every i. Using again that Γ is a DCC set, we see that after passing to a subsequence r + s times, we may assume that each of the sequences (pi,j )i≥1 and (qi,j )i≥1 is non-decreasing. Recall that ci ≤ N/pi,j and 1 ≤ N/qi,j for every i and j, hence the sequences (pi,j )i≥1 and (qi,j )i≥1 are bounded. We put pj = limi→∞ pi,j and qj = limi→∞ qi,j . Fs Fr (j) (j) Let cDi := lct(Wi ,B" ) (ADi ), where ADi = j=1 (ai )pj and BDi = j=1 (bi )qj . Since pi,j ≤ pj and qi,j ≤ qj for every i and j, it follows that cDi ≤ ci for every i. On the other hand, for every η ∈ (0, 1), we have pi,j /pj , qi,j /qj > 1−η for all j and all i D 0. This implies ci ≤ cDi /(1 − η) for all i D 0. We deduce that the sequence (cDi )i≥1 contains a strictly increasing subsequence converging to c. Therefore, in order to derive a contradiction, we may assume that pi,j = pj and qi,j = qj for every i and j. Case 1. We ﬁrst treat the case when Bi = OX for every i. The argument for this case now follows closely the proof of [3, Theorem 5.1], using though the version of generic limit construction introduced in the previous section. We consider the sequences of ideals (r)

(1)

(pi )i∈I0 , (ai )i∈I0 , . . . , (ai )i∈I0 , (m)i∈I0 , where m is the maximal ideal in R. We choose a generic limit (p, a(1) , . . . , a(r) , mK ) constructed as in §4, with a(j) proper ideals in RK = K[[x1 , . . . , xN ]]. As in Proposition 3.3, we consider the scheme W = Spec(RK /pRK ), and the ideals a(j) = a(j) /(pRK ). Note that by Remark 3.5 all a(j) are nonzero. Let A be Fr the R-ideal j=1 (a(j) )pj . It follows from Corollary 3.4 that c = lct(W, A). By Lemma 2.9, we are able to ﬁnd a nonnegative real number t such that lct(W, mtK ·A) = c, and this is computed by a divisor E over W having center equal to the closed point (recall that mK deﬁnes the closed point of W , and mi deﬁnes the closed point on Wi ). It follows from Proposition 3.3 iii) that we can ﬁnd dE such that for every d ≥ dE the following holds: there is an inﬁnite subset IdE ⊆ I0 such that for each i ∈ IdE there is a divisor Ei over Wi having center equal to the Fr (j) closed point, computing the log canonical threshold lct(Wi , mti · j=1 (ai +mdi )pj ), (j)

and such that ordE (a(j) + mdK ) = ordEi (ai + mdi ) for every j. Fix d ≥ dE such that, in addition, d > ordE (a(j) ) for all j ≥ 1. Then, because lct(W, mtK · A) is computed by E, and a(j) ⊆ a(j) + mdK , it follows that F lct(W, mtK · A) = lct(W, mtK · j (a(j) + mdK )pj ), (10)

and the right-hand side is computed by E. On the other hand, it follows from Proposition 3.3 ii) that we may assume that for all i ∈ IdE G 7 G 7 F F (j) lct W, mtK · j (a(j) + mdK )pj = lct Wi , mti · j (ai + mdi )pj .

(11)

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Since d > ordE (a(j) ), we have (j)

ordE (a(j) ) = ordE (a(j) + mdK ) = ordEi (ai + mdi ) < d for every i ∈ IdE , and every j ≥ 1. It follows from Proposition 2.20 that 7 G 7 F (j) G F (j) lct Wi , mti · j (ai )pj = lct Wi , mti · j (ai + mdi )pj .

(12)

(13)

By combining equations (10), (11) and (13), we conclude that 7 F (j) G c = lct Wi , mt · j (ai )pj ≤ lct(Wi , Ai ) < c, a contradiction. This completes the proof of this case. Case 2. We now treat the general case. Consider the sequences of ideals (1)

(r)

(1)

(s)

(pi )i∈I0 , (ai )i∈I0 , . . . , (ai )i∈I0 , (bi )i∈I0 , . . . , (bi )i∈I0 . Again, we construct a generic limit (p, a(1) , . . . , a(r) , b(1) , . . . , b(s) ) as in §3. Let (j)

W = Spec(RK /pRK ), and a(j) = a(j) /pRK and b = b(j) /pRK . We consider the Fs Fr (j) R-ideals A = j=1 (a(j) )pj and B = j=1 (b )qj . " For every cD < c, we have ci > cD for i D 0. Therefore lct(Wi , Bi · Aci ) ≥ 1 " for such i. By Proposition 3.3 ii), lct(W, B · Ac ) is a limit point of the sequence " " (lct(Wi , Bi · Aci ))i≥1 , hence lct(W, B · Ac ) ≥ 1. Since this holds for every cD < c, we have lct(W, B · Ac ) ≥ 1. Another application of Proposition 3.3 ii) gives that lct(W, B · Ac ) is a limit point of the sequence (lct(Wi , Bi ·Aci ))i≥1 . On the other hand, it follows from Case 1 that the set {lct(Wi , Bi ·Aci ) | i ≥ 1} contains no strictly increasing sequences. We deduce that there are inﬁnitely many i such that lct(Wi , Bi ·Aci ) ≥ lct(W, B·Ac ) ≥ 1. For every such i we have ci ≥ c, a contradiction. This completes the proof of the theorem. Remark 4.3. It follows from Proposition 2.11 that the statement in the above theorem implies the version stated in the Introduction in Theorem 1.1 in terms of bounded families of singularities.

A. Sheaves of diﬀerentials for schemes of ﬁnite type over a formal power series ring In this appendix we work in the following setting. Let k be a ﬁeld of characteristic zero, and R = k[[x1 , . . . , xn ]]. All our schemes will be of ﬁnite type over such a formal power series ring. We note that since R is an excellent ring (see [15, p. 260]), it follows that the nonsingular locus of such a scheme is open. Furthermore, R is universally catenary. The usual sheaves of diﬀerentials over k are not the right objects in our setting (in particular, they are not coherent). Our aim in this section is to introduce an appropriate version of sheaves of diﬀerentials that is better behaved.

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For every R-module M , the special k-derivations %n ∂f D : R → M are those kderivations with the property that D(f ) = i=1 ∂xi D(xi ) for every f ∈ R. Note that this is automatically true for a k-derivation if M is separated in the (x1 , . . . , xn )-adic topology, but not in general. For an R-algebra A and an A-module M , the module DerDk (A, M ) of special kderivations consists of all k-derivations D : A → M such that the restriction of D to R is a special k-derivation R → M . It is clear that DerDk (A, M ) is an A-submodule of Derk (A, M ). Note that the deﬁnition depends on the ﬁxed ring R. If w : M → N is a morphism of A-modules, composing with w induces a morphism of A-modules DerDk (A, M ) → DerDk (A, N ). We say that A has a module of special diﬀerentials if there is an A-module ΩDA/k with a special k-derivation dDA/k : A → ΩDA/k that induces an isomorphism of A-modules HomA (ΩDA/k , M ) → DerDk (A, M ) for every A-module M . Of course, in this case ΩDA/k is called the module of special diﬀerentials (note that it is unique, up to a canonical isomorphism commuting with dDA/k ). In order to avoid cluttering the notation we do not include R in the notation for ΩDA/k . However, the reader should keep in mind that the deﬁnition was made in reference to a ﬁxed R. Lemma A.1. If A → B = A/I is a surjective morphism of R-algebras, and if ΩDA/k exists, then ΩDB/k exists, and we have an exact sequence δ

u

I/I 2 → ΩDA/k ⊗A B → ΩDB/k → 0, where δ(f ) = dDA/k (f ) ⊗ 1 and u(dDA/k (f ) ⊗ 1) = dDB/k (f ). Proof. The assertion follows as in the case of usual diﬀerentials from the fact that the corresponding sequence of B-modules 0 → DerDk (B, M ) → DerDk (A, M ) → HomA (I/I 2 , M ) is exact for every A-module M . Lemma A.2. The module of special diﬀerentials ΩDR/k exists, and it is a free R-module with basis dDR/k (x1 ), . . . , dDR/k (xn ). Proof. The assertion follows from the fact that by deﬁnition, every D ∈ DerDk (R, M ) is uniquely determined by the D(xi ), which can be chosen arbitrarily. Lemma A.3. Let S be an R-algebra, and A = S[yi | i ∈ I] a polynomial ring over S. If ΩDS/k exists, then ΩDA/k exists and it is isomorphic to the direct sum of ΩDS/k ⊗S A and a free A-module with basis {dDA/k (yi ) | i ∈ I}. Proof. The assertion follows from the fact that every D ∈ DerDk (A, M ) is uniquely determined by D|S and by the D(yi ), which can be chosen arbitrarily.

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By combining the above three lemmas we deduce the following existence result. Corollary A.4. For every R-algebra A, the module ΩDA/k exists. Furthermore, if A is of ﬁnite type over R, then ΩDA/k is ﬁnitely generated over A. Remark A.5. Since DerDk (A, M ) ⊆ Derk (A, M ) for every A-module M , it follows that we have a surjective morphism ΩA/k → ΩDA/k . In particular, ΩDA/k is generated as an A-module by {dDA/k (a) | a ∈ A}. Lemma A.6. If ϕ : A → B is a morphism of R-algebras, then we have an exact sequence v u ΩDA/k ⊗A B → ΩDB/k → ΩB/A → 0, where u(dDA/k (f ) ⊗ 1) = dDB/k (f ) and v(dDB/k (f )) = dB/A (f ). Proof. The assertion follows as in the case of usual derivations from the fact that for every B-module M , the corresponding sequence 0 → DerA (B, M ) → DerDk (B, M ) → DerDk (A, M ) is exact (note that if D ∈ DerA (B, M ), then D is trivially a special k-derivation, since its restriction to R is zero). Lemma A.7. Let A be an R-algebra. If S is a multiplicative system in A, then we have a canonical isomorphism S −1 ΩDA/k @ ΩDS −1 A/k . Proof. It is enough to note that for every S −1 A-module M , we have canonical isomorphisms HomS −1 A (S −1 ΩDA/k , M ) @ HomA (ΩDA/k , M ) @ DerDk (A, M ) @ DerDk (S −1 A, M ). (14) The last isomorphism follows from the fact that every k-derivation D : A → M ) : S −1 A → M , and it is clear that D is special if and admits a unique extension D ) only if D is. The assertion in the lemma follows from (14). The case when A is regular will play an important role. We show that in this case ΩDA/k is locally free. In fact, we have the following more precise result. Proposition A.8. If A is an algebra of ﬁnite type over R, and if q ∈ Spec(A) is such that Aq is a regular ring of dimension r, then ΩDAq /k is a free Aq -module, of rank equal to r + dimk(q) (ΩDk(q)/k ), where k(q) = Aq /qAq . Furthermore, if u1 , . . . , ur ∈ A induce a regular system of parameters in Aq , then the images of dDA/k (u1 ), . . . , dDA/k (ur ) in ΩDAq /k are part of a basis. Proof. Our argument is based on the results in [16]. Since the regular locus of A is open, we may replace A by a localization Af in order to assume that A is a regular ring, and further, that it is a domain. Let us choose an isomorphism A @ S/P , with S = R[y1 , . . . , yN ] and P ∈ Spec(S). Let Q ∈ Spec(S) be such that q = Q/P . We put s = codim(P ), so that codim(Q) = r + s.

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A key property of S is that it satisﬁes the strong Jacobian condition over k. At the prime Q, this means that there are D1 , . . . , Dr+s ∈ Derk (S) · SQ , and %r+s w1 , . . . , wr+s ∈ Q, such that det(Di (wj )) ∈ K QSQ and [Di , Dj ] ∈ \=1 SQ · D\ . The fact that S satisﬁes this condition follows from [16, Theorem 6], which says that rings with the strong Jacobian condition are closed under taking polynomial or formal power series rings (the statement therein is in terms of absolute derivations, but the proof goes through if one works with k-derivations). In this case [16, Theorem 5] implies that SQ /P SQ is regular if and only if there are w1D , . . . , wsD ∈ P such that some s-minor of the matrix (Di (wjD ))i,j does not lie in QSQ . Lemma A.1 gives (after localizing and using Lemma A.7) an exact sequence of Aq -modules ϕ P SQ /P 2 SQ → ΩDS/k ⊗S Aq → ΩDAq /k → 0. Since SQ /P SQ is regular, P SQ /P 2 SQ is a free Aq -module of rank s. Note that since S is separated with respect to the (x1 , . . . , xn )-adic topology, we have DerDk (S) = Derk (S). Therefore D1 , . . . , Dr+s deﬁne an Aq -linear map ψ : ΩDS/k ⊗S Aq → Ar+s q such that ψ ◦ ϕ is split injective. It follows from the above exact sequence that ΩDAq /k is a free Aq -module of rank n + N − s, and we also see that w1D , . . . , wsD generate P SQ . Running the same argument with P replaced by Q, we see that dimk(q) (ΩDk(q)/k ) = n + N − (r + s), which gives the assertion about the rank of ΩDAq /k . For the last assertion in the proposition, note that if the u )i ∈ S are lifts of the ui , then it follows that u )1 , . . . , u )r , w1D , . . . , wsD give a minimal system of generators of QSQ . By writing w1 , . . . , wr+s in terms of this system of generators, we see that dDS/k () u1 ) ⊗ 1, . . . , dDS/k () ur ) ⊗ 1, dDS/k (w1D ) ⊗ 1, . . . , dDS/k (wsD ) ⊗ 1 are part of a basis of ΩDS/k ⊗S Aq . We deduce from this the last assertion in the proposition. For future reference, we include here the following lemma, describing the behavior of the module of special diﬀerentials with respect to ﬁeld extensions. Lemma A.9. Let K n→ L be a ﬁeld extension, where K is an R-algebra. In this case we have an exact sequence 0 → ΩDK/k ⊗K L → ΩDL/k → ΩL/K → 0. Proof. By Lemma A.6, it is enough to show that the morphism ΩDK/k ⊗K L → ΩDL/k is injective; equivalently, for every L-module M , the map DerDk (L, M ) → DerDk (K, M ) is surjective. This follows from the fact that the map Derk (L, M ) → Derk (K, M ) is surjective (recall that char(k) = 0), by noticing that a k-derivation D : L → M is special if and only if D|K is special. We will need a comparison between the usual module of diﬀerentials for schemes of ﬁnite type over a ﬁeld, and the module of special diﬀerentials for the completion at a closed point. We consider the following setting. Suppose that ϕ : A → B is

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a morphism of ﬁnitely generated k-algebras, and m is a maximal ideal in A. The Im and B D = B ⊗A A Im . ﬁeld K = A/m is a ﬁnite extension of k. We put AD = A By Cohen’s Structure Theorem, we can ﬁnd a surjective local morphism of kalgebras ψ : S = K[[x1 , . . . , xN ]] → AD . In this case we take R = k[[x1 , . . . , xN ]] and mR = (x1 , . . . , xN ). Note that AD becomes naturally an R-algebra via the inclusion R n→ S. We use this structure when considering the modules of special derivations ΩDA" /k and ΩDB " /k . Since K/k is ﬁnite, AD is ﬁnite as an R-algebra, hence B D is a ﬁnitely generated R-algebra. Proposition A.10. With the above notation, we have a canonical isomorphism ΩB/k ⊗B B D @ ΩDB " /k .

(15)

Proof. We ﬁrst show that ΩA/k ⊗A AD @ ΩDA" /k . Note that if M is a ﬁnitely generated AD -module, then M is separated with respect to the mA" -adic topology (where mA" is the maximal ideal of AD ), hence it is separated with respect to the mR -adic topology. Therefore DerDk (AD , M ) = Derk (AD , M ). Let dA/k : A → ΩA/k be the universal k-derivation on A, and let j : ΩA/k → ΩA/k ⊗A AD be the canonical map. Note that ΩA/k ⊗A AD is complete in the mAm -adic topology, hence we get a D D D 4 unique d4 A/k ∈ Derk (A , ΩA/k ⊗A A ) such that dA/k ◦ι = j◦dA/k , where ι : A → A is D D the completion map. Since ΩA/k ⊗A A is a ﬁnitely generated A -module, it follows that d4 A/k is a special derivation, and we have a unique morphism of A-modules

f : ΩDA" /k → ΩA/k ⊗A AD such that f ◦ dDA" /k = d4 A/k . D On the other hand, since dA" /k is a derivation, there is a unique morphism of A-modules g : ΩA/k → ΩDA" /k such that g ◦ dA/k = dDA/k ◦ ι. Since ΩDA" /k is ﬁnitely generated over AD by Corollary A.4, it is complete, hence g induces a (unique) D D 4 morphism of AD -modules g# : Ω # ◦ j = g. It A/k = ΩA/k ⊗A A → ΩA" /k such that g is now easy to check that f and g# are inverse isomorphisms. In order to prove the general statement for B, let us write B @ A[y1 , . . . , ym ]/I. In this case we have m Q 3 . ΩB/k @ (ΩA/k ⊗A B) ⊕ B · dB/k (yi ) /B · {dB/k (u) | u ∈ I}, (16) i=1 D

which after tensoring with B gives a description of ΩB/k ⊗B B D . Since we have a corresponding isomorphism B D @ AD [y1 , . . . , ym ]/(ι(I)), using Lemmas A.1 and A.3 we get an analogous formula for ΩDB " /k . The isomorphism (15) now follows from the corresponding isomorphism in the case B = A. It is standard to deduce from Lemma A.7 that for every scheme X over R, there is a quasicoherent sheaf ΩDX/k such that for every aﬃne open subset U of X, the restriction of ΩDX/k to U is canonically isomorphic to the sheaf associated to ΩDO(U )/k . It follows from Corollary A.4 that if X is of ﬁnite type over R, then ΩDX/k is coherent. Furthermore, Proposition A.8 implies that if X is nonsingular, then ΩDX/k is locally free. The exact sequences in Lemmas A.1 and A.6 globalize in a straightforward way.

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We now use the sheaves of special diﬀerentials to introduce the notion of relative canonical class in this setting. Let X be a normal scheme of ﬁnite type over R. Since the discussion that follows can be done separately on each connected component of X, we may and will assume that X is irreducible. Recall that since R is excellent, the nonsingular locus Xreg of X is an open subset of X. Since X is normal, the complement X 1 Xreg has codimension ≥ 2 in X. In particular, restriction induces an isomorphism of class groups Cl(X) @ Cl(Xreg ). The restriction ΩDX/k |Xreg is locally free, and let M be its rank. On X we have a Weil divisor KX , uniquely deﬁned up to rational equivalence, such that O(KX )|Xreg @ ∧M ΩDXreg /k . As in the case of schemes of ﬁnite type over a ﬁeld, we say that X is Q-Gorenstein if there is a positive integer r such that rKX is a Cartier divisor (the smallest such r is the index of X; any other r with this property is a multiple of the index). Suppose now that π : Y → X is a proper birational morphism of schemes over R, with Y nonsingular. The following lemma shows that the relative canonical class KY /X can be deﬁned in the same way as in the case of schemes of ﬁnite type over a ﬁeld (see [11]). Lemma A.11. With the above notation, the following hold: i) We may take KX = π∗ (KY ). ii) If rKX is Cartier, then there is a unique Q-divisor KY /X supported on the exceptional locus of π such that rKY and π ∗ (rKX ) + rKY /X are linearly equivalent. If X is nonsingular, then KY /X is eﬀective and its support is the exceptional locus Exc(π). iii) Suppose that X is nonsingular, and that E1 + · · · + Eq is a divisor on X having simple normal crossings. If F is a prime nonsingular divisor on Y with corresponding valuation ordF , and if ordF (Ei ) = ai for every i, then ordF (KY /X ) ≥ a1 + · · · + aq − 1. Proof. In order to prove i), we may restrict to Xreg , and therefore assume that X is nonsingular. If y ∈ Y and x = π(y), then Lemma A.6 gives an exact sequence w

U := ΩDX/k,x ⊗ OY,y → V := ΩDY /k,y → ΩY /X,y → 0. Since π is birational, it follows from the Dimension Formula (see [15, Theorem 15.6]) that dim(OY,y ) = dim(OX,x ) + trdeg(k(y)/k(x)). Since dimk(y) (Ωk(y)/k(x) ) = trdeg(k(y)/k(x)), we deduce from Lemma A.9 and Proposition A.8 that U and V are free OY,y -modules of the same rank M . It follows that ∧M w is given by the equation of an eﬀective divisor KY /X . The support of this divisor is the locus where π is not ´etale, which in this case is precisely the exceptional locus of π. The assertions in i) and ii) now easily follow. Due to the last assertion in Proposition A.8, we can deduce iii) via the same computation as in the usual case of schemes of ﬁnite type over a ﬁeld.

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Remark A.12. It follows from the above proof that if X is nonsingular, then KY /X is independent of the structure of X as an R-scheme. Indeed, KY /X is the eﬀective divisor deﬁned by the 0th Fitting ideal of ΩY /X . It is not clear to us whether the same remains true if X is singular. ϕ

π

Lemma A.13. If Y D → Y → X are proper birational morphisms, with both Y and Y D nonsingular, and if X is Q-Gorenstein, then KY " /X = KY " /Y + ϕ∗ (KY /X ).

(17)

Proof. It is enough to observe that if rKX is Cartier, then r(KY " /Y + ϕ∗ (KY /X )) is (π ◦ ϕ)-exceptional, and it is linearly equivalent to rKY " − (π ◦ ϕ)∗ (rKX ). In the next proposition we consider an integral scheme X, of ﬁnite type over a ﬁeld k (assumed, as always, to have characteristic zero). Suppose that π : Y → X is a proper birational morphism, with Y nonsingular. Let x ∈ X be a closed point, and consider the Cartesian diagram W

h

!Y π

f

# 4 Z = Spec(O X,x )

g

# !X

(see Remark 2.10 for general properties of such a diagram). From now on, let us assume that X is normal. In this case W is connected: otherwise the ﬁber over the unique closed point of Z would be disconnected, but this is the same as the ﬁber π −1 (x). Since π is proper and birational, we deduce that f as well has these two properties. We consider both Z and W as schemes over a formal power series ring over k, as in Proposition A.10. Since g and h are ﬂat, we may pull back Weil divisors via both g and h. Proposition A.14. With the above notation, we may take KZ = g ∗ (KX ). In particular, rKX is Cartier in a neighborhood of x if and only if rKZ is Cartier, and in this case h∗ (KY /X ) = KW/Z . Proof. Since g is a regular morphism, it follows that g −1 (Xreg ) = Zreg . The ﬁrst assertion in the proposition follows from the fact that if g0 : Zreg → Xreg is the restriction of g, then g0∗ (ΩXreg /k ) @ ΩDZreg by Proposition A.10. Furthermore, the same proposition implies that we may take KW = h∗ (KY ). Note now that if D is a divisor on X, then D is Cartier in a neighborhood of x if and only if g ∗ (D) is Cartier. Indeed, for this we may assume that D is eﬀective, 4 and let I = O(−D) · OX,x . In this case O(−g ∗ (D)) = I · O X,x , since for every 4 prime ideal P in OX,x and every minimal prime ideal Q in O X,x containing P , we 4 4 have P · (OX,x )Q = Q · (OX,x )Q (this follows from the fact that the ﬁber over P 4 is nonsingular). It is now enough to note that I · O X,x is principal if and only if

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4 I is principal (more generally, I and I · O X,x have the same minimal number of generators). In particular, we see that rKX is Cartier in a neighborhood of x if and only if rKZ is Cartier. The last assertion in the proposition now follows from the fact that h∗ (KY /X ) is supported on the inverse image via h of the exceptional locus of π, hence on the exceptional locus of f . Remark A.15. Suppose that F is a prime nonsingular divisor on Y . The pull-back h∗ (F ) is a nonsingular divisor on W . If we consider the irreducible components E1 , . . . , Em of h∗ (F ), then the restriction of each ordEi to the function ﬁeld of X is equal to ordF . We note that if the center of F is x, then h∗ (F ) is abstractly isomorphic to F . In particular, h∗ (F ) is a prime divisor.

B. Rational Q-Gorenstein singularities in families Throughout this appendix, all varieties and schemes are of ﬁnite type over a ﬁeld k of characteristic zero. At one point, we will need to assume that k = C. Our goal is to prove Theorem B.8 on the behavior of the canonical class and Gorenstein index in families. This implies the corollary about the generic behavior of the log canonical threshold in families that is used in the proof of Proposition 3.3. In fact, Theorem B.8 follows from the following more precise result, for which we need to assume that k is the ﬁeld of complex numbers. Given a positive integer r, we say that a normal variety X is r-Gorenstein at a point x if rKX is Cartier at x. For a scheme X → T over T we denote by Xξ the ﬁber over the not necessarily closed point ξ ∈ T . Theorem B.1. Let f : X → T be a morphism of normal varieties over C such that every ﬁber of f is normal. Then there are a positive integer s and a nonempty Zariski open set T ◦ ⊆ T such that for every closed point t ∈ T ◦ , if Xt has rational singularities at a closed point x, then the following conditions are equivalent: (a) Xt is Q-Gorenstein at x; (b) Xt is s-Gorenstein at x; (c) X is Q-Gorenstein at x; (d) X is s-Gorenstein at x. Before giving the proof of the theorem, we start with some general considerations. Recall ﬁrst Grothendieck’s Generic Freeness Theorem (see, for example, [5, Theorem 14.4]). Theorem B.2 (Generic Freeness Theorem). Let ϕ : A → B be a ring homomorphism of ﬁnite type, with A a Noetherian integral domain. If M is a ﬁnitely generated B-module, then there is a nonzero a ∈ A such that Ma is a free Aa -module.

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Corollary B.3. If f : X → T is a scheme morphism of ﬁnite type, with T a Noetherian integral scheme, then there is a nonempty open subset W in T such that f −1 (W ) → W is ﬂat. Furthermore, given a complex of coherent sheaves on X C : F D → F → F DD , with homology sheaf H(C), we can choose W such that for every ξ ∈ W the canonical morphism H(C) ⊗ OXξ → H(C ⊗ OXξ ) is an isomorphism. Proof. The ﬁrst assertion follows easily from the theorem. For the second one, note that by the theorem, we may choose W such that the images and the cokernels of the arrows in C are all ﬂat over W . It is then easy to see that W has the required property. The following lemmas will be used in the proof of Theorems B.1 and B.8. Lemma B.4. Let f : X → T be a morphism of normal schemes such that all ﬁbers of f are normal. For every positive integer m, there is an open subset Wm ⊆ T such that for every ξ ∈ Wm we have a canonical isomorphism O(mKX )|Xξ ∼ = O(mKXξ ). In particular, for every ξ ∈ Wm , the divisor mKX is Cartier at a point x ∈ Xξ if and only if mKXξ is Cartier at x. Proof. By Corollary B.3, after replacing T by an open subset we may assume that f is ﬂat. We may clearly also assume that T is nonsingular. In particular, if x is a nonsingular point of Xξ , then both f and X are smooth at x. In this case we clearly have a canonical isomorphism O(mKX )|Xξ ∼ = O(mKXξ ) in a neighborhood of x (where Xξ is considered as a scheme over Spec(k(ξ)). Since the complement of (Xξ )reg in Xξ has codimension ≥ 2, it is enough to ﬁnd Wm such that for every ξ ∈ Wm , the restriction O(mKX )|Xξ is reﬂexive. After covering X by aﬃne open subsets, we may assume that X is aﬃne. Since O(mKX ) is reﬂexive, we may write it as the kernel of a morphism ϕ : E1 → E0 of free coherent sheaves on X. By Corollary B.3, there is an open subset Wm ⊆ T such that for every ξ in Wm the restricted sheaf O(mKX )|Xξ is isomorphic to the kernel of the restriction of ϕ to Xξ , which is a reﬂexive sheaf. This completes the proof. Lemma B.5. If X is a normal scheme, and U = {x ∈ X | X is Q-Gorenstein at x} is the Q-Gorenstein locus of X, then U is open in X, and there is a positive integer s such that sKX is Cartier on U . Proof. For every positive integer m, the set Um = {x ∈ X | X is m-Gorenstein at x}

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: is open in X (it is nonempty, since it contains Xreg ). Note that U = m≥1 Um . Furthermore, we have Uk ⊆ Um if k divides m. It follows by the Noetherian property that there is a unique maximal set among all these open sets. In other words, there is a positive integer s such that U = Us . Lemma B.6. Let X be a normal scheme, and let g : Y → X be a resolution of singularities. For a positive integer m, the divisor mKX is Cartier at a closed point x ∈ X if and only if there is an open neighborhood V of x and a g-exceptional divisor E on Y such that O(mKY ) ∼ = O(E) on g −1 (V ). Furthermore, if the ground ﬁeld is C, then it is enough to ﬁnd an open neighborhood V of x in the analytic topology such that O(mKY )an ∼ = O(E)an on g −1 (V ). Proof. Note that after ﬁxing the Cartier divisor KY on Y , we may take KX = g∗ KY . If mKX is Cartier, then mKY −g ∗ (mKX ) is an integral exceptional divisor. Thus, given x ∈ X such that mKX is Cartier at x, it is enough to take an open neighborhood V of x where mKX is principal. Conversely, if there is an E as in the statement, then taking the push-forward and observing that g∗ (mKY ) = mKX and g∗ E = 0, we see that mKX is linearly equivalent to zero in a neighborhood of x. Suppose now that X is a complex variety, and assume that O(mKY )an ∼ = O(E)an on g −1 (V ), where V is an open neighborhood of x in the analytic topology. It follows that there is a meromorphic function ϕ on Y such that divY (ϕ) = mKY − E on g −1 (V ). In this case divX (ϕ) = mKX on V . Therefore O(mKX )an is locally free of rank one at x. Since O(mKX ) and O(mKX )an have isomorphic completions at x, it follows by Nakayama’s Lemma that O(mKX ) is locally free of rank one at x, hence mKX is Cartier at this point. We are now ready to prove the key result of this appendix. Proof of Theorem B.1. In this proof we only consider the closed points of the schemes involved. Let g : Y → X be a resolution of singularities whose exceptional locus is a divisor with simple normal crossings. 8 By a theorem of Verdier [20], we can write X as a ﬁnite disjoint union X = X α , with each X α an irreducible locally closed subset of X, such that the restriction g α : Y α → X α of g to Yα = g −1 (Xα ) is topologically locally trivial. Let Z α := X α 1 X α (the closure being taken inside X). Note that each Z α is a closed subset of X. By Lemma B.5, there is a positive integer s such that X is Q-Gorenstein at a point x if and only if X is s-Gorenstein at x. By generic smoothness, generic ﬂatness, and Lemma B.4, after possibly replacing T by a nonempty open subset, we can assume that the following properties hold: (1) T is smooth; (2) Y → T is smooth, the exceptional locus of g has relative simple normal crossings over T , and for every point t ∈ T , the induced morphism gt : Yt → Xt is a resolution of singularities and every gt -exceptional divisor is the restriction to Yt of a g-exceptional divisor;

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(3) X is ﬂat over T , and both X α and Z α are ﬂat over T for every α; (4) For every t ∈ T , there is a canonical isomorphism O(sKX )|Xt ∼ = O(sKXt ). We will show that after this reduction the conclusion of the theorem holds for every t ∈ T. Fix any t ∈ T , and suppose that x is a point where Xt has rational singularities. Since f is ﬂat and T is smooth, this implies that X has rational singularities at x, and hence in a neighborhood of x (cf. [6, Th´eor`eme 2 and Th´eor`eme 4]). By Lemma B.5, we see that the conditions (c) and (d) are equivalent. Furthermore, condition (4) implies that (b) and (d) are equivalent, and clearly (b) implies (a). Therefore, in order to conclude it suﬃces to show that (a) implies (c). We thus assume that (a) holds, that is, that there is a positive integer m such that mKXt is Cartier at x. We denote by Xtα , (Xα )t , and Ztα the ﬁbers of X α , Xα , and Z α over t. Let A := {α | x ∈ Xtα }. Note that Q K 3 x ∈ Int Xtα . α∈A

Indeed, if this were false, then for every open neighborhood V of x in Xt we could ﬁnd an α ∈ K A such that V ∩ Xtα K= ∅. By considering a nested sequence of open neighborhoods of x, we see that we can pick α independent of V . As this holds for every V , we conclude that x ∈ Xtα , which contradicts the deﬁnition of A. Claim. We have Q K 3 x ∈ Int Xα . α∈A

Proof of claim. 8 We argue by contradiction. Let us assume that x is not in the interior of α∈A X α . Arguing as above, we conclude that there is an α ∈ K A such α α α that x ∈ X . Consider the morphism X → T . We have x ∈ (X )t , and since x∈ K Xtα , there is an open neighborhood V of x in (X α )t that is disjoint from Xtα , and hence from X α . Therefore V is contained in Z α , hence in Ztα . The closure of V in (X α )t , and hence the closure of Ztα in (X α )t , contains some irreducible component W of (X α )t . Since both Z α and X α are ﬂat over T , it follows that if xD ∈ W is a general (closed) point, then dim(W ) = dim(OZ α ,x" ) − dim(T ) = dim(OX α ,x" ) − dim(T ) (see [8, Proposition III.9.5]). The fact that dim(OZ α ,x" ) = dim(OX α ,x" ) contradicts the fact that Z α is a proper closed subset of the irreducible set X α , and thus completes the proof of the claim. We now ﬁx 8 a small contractible analytic open neighborhood V ⊆ X of x fully contained in α∈A X α , and such that H 1 (V, OVan ) = 0. Let Vt = V ∩ Xt . We may and will assume that V has rational singularities. Furthermore, we may assume that Vt is contained in any given neighborhood of x, hence by Lemma B.6 and

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the fact that mKXt is Cartier at x we may assume that there is a gt -exceptional divisor Et on Yt such that O(mKYt )an ∼ = O(Et )an

on gt−1 (Vt ).

(18)

It follows from condition (2) that mKYt is the restriction of mKY to Yt , and Et is the restriction of a g-exceptional divisor E on Y . By Lemma B.6, in order to conclude the proof of the theorem, it suﬃces to show that there is an ] ≥ 1 such that L\ is trivial, where L = O(mKY − E)an |g−1 (V ) . Let γ = c1 (L) ∈ H 2 (g −1 (V ), Z). For every x ∈ V , we denote by γx the image of γ in H 2 (Yx , Z) via the map induced by g −1 (x) = Yx n→ g −1 (V ). Arguing by contradiction, let us assume that L\ is nontrivial for all ] ≥ 1. It follows from [13, (12.1.4)] and the proof therein that in this case we can ﬁnd a g-exceptional curve C ⊂ Y , with image p := g(C) in V , such that (L · C) = K 0. In particular, γp K= 0. By our choice of V , we have p ∈ X α for some α ∈ A. Note that Vt ∩ Xtα K= ∅ by the deﬁnition of A (recall that by [17, Proposition 5], the analytic closure of Xtα in Xt agrees with the Zariski closure Xtα ). Pick any point q ∈ Vt ∩ Xtα . Since X α is connected, and hence path connected, we can ﬁx a path w : [0, 1] → X α joining p to q. As g α is topologically locally trivial, moving along the path w induces an isomorphism H 2 (Yp , Z) ∼ = H 2 (Yq , Z). Note that γp is mapped to γq via this isomorphism, hence γq = K 0. On the other hand, (18) implies that L|g−1 (V )∩Yt is trivial, hence so is L|Yq . Therefore γq = 0, a contradiction. This completes the proof of the theorem. Remark B.7. It follows from the above proof that in Theorem B.1 one can take any s as given by Lemma B.5, that is, such that sKX is Cartier on the largest open subset of X that is normal and Q-Cartier. We will need the following version of the result, which holds over any algebraically closed ﬁeld k of characteristic zero. Theorem B.8. Let f : X → T be a morphism of schemes of ﬁnite type over k, with T integral, and let σ : T → X be a section of f , i.e. f ◦ σ = 1T . Suppose that we have a countable dense subset T0 ⊆ T of closed points such that for all t ∈ T0 , at σ(t) the ﬁber Xt := f −1 (t) is Q-Gorenstein and has rational singularities. Then there is a nonempty open subset U of T , and a positive integer s such that i) X is normal and sKX is Cartier in a neighborhood of σ(U ). ii) For every (not necessarily closed) point ξ ∈ U , the ﬁber Xξ is normal at σ(ξ), and we have a canonical isomorphism O(sKX )|Xξ @ O(sKXξ ) in a neighborhood of σ(ξ). In particular, sKXξ is Cartier at σ(ξ).

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Proof. It is clear that we may replace T by any nonempty open subset V (note that the set T0 ∩ V is countable and dense in V ). Furthermore, if W ⊆ X is an open subset such that σ −1 (W ) is nonempty, it is enough to prove the theorem for W ∩ f −1 (σ −1 (W )) → σ −1 (W ) (note that σ induces a section of this morphism). After replacing T by an open smooth subset, we may assume that T is smooth, and f is ﬂat (we again use generic ﬂatness). By [6, Th´eor`eme 4], there is an open subset W1 of X whose closed points x ∈ W1 are precisely those such that Xf (x) has rational singularities at x. Since σ(t) ∈ W1 for every t ∈ T0 , we see that σ −1 (W1 ) is nonempty. After replacing X by W1 ∩ f −1 (σ −1 (W1 )), we may assume that every closed ﬁber Xt has rational singularities. In this case, by [6, Th´eor`eme 2], X has rational singularities. Furthermore, all ﬁbers of f are normal. We apply Lemma B.5 to get the open subset W ⊆ X consisting of the points in X where KX is Q-Cartier. Let s be such that sKX is Cartier on W . In order to prove the theorem it is enough to show that σ −1 (W ) is nonempty. Indeed, if this is the case we may replace X by W ∩ f −1 (σ −1 (W ), in which case sKX is Cartier. After replacing T by a nonempty open subset, we may assume by Lemma B.4 that O(sKX )|Xξ @ O(sKXξ ) for every ξ ∈ T . This would prove the theorem. If the ground ﬁeld is C, then by Theorem B.1 (see also Remark B.7) we may replace T by an open subset and assume that for every closed point t ∈ T , the divisor sKXt is Cartier at a closed point x if and only if x ∈ W . In this case we see that T0 ⊆ σ −1 (W ), hence σ −1 (W ) is nonempty. For an arbitrary k, we can ﬁnd a subﬁeld k D of k of countable transcendence degree over Q, such that there are morphisms f D : X D → T D and σ D : T D → X D of schemes over k D , with f and σ obtained from f D , respectively σ D , by base extension via Spec(k) → Spec(k D ), and such that the points in T0 are deﬁned over k D . If ϕ : T → T D is the natural morphism, it follows that T0D := ϕ(T0 ) consists of k D rational closed points. ) → T) and σ ) be the There is an embedding k D n→ C. Let f): X ) : T) → X D D morphisms obtained from f , respectively σ , by base extension to C. If ψ : T) → T D is the natural morphism, we choose (closed) points ) t ∈ ψ −1 (tD ) for all tD ∈ T0D . Let ) T0 be the set consisting of these closed points. It follows from Lemma 2.14 i) +0 , the ﬁber X )e is Q-Gorenstein and with rational singularities that for every ) t∈T t at σ )() t) (the assertion about rational singularities follows easily from deﬁnition by considering base extensions of resolutions of singularities). + ⊆ X ) are the subsets where X D and On the other hand, if W D ⊆ X D and W ) X, respectively, are normal and Q-Gorenstein, then by Lemma 2.14 i) we see that + = W D ×Spec(k" ) Spec(C) and W = W D ×Spec(k" ) Spec(k). Furthermore, the W + , respectively. We have already divisors sKX " and sKXe are Cartier on W D and W −1 + + )) seen that σ ) (W ) is a nonempty open subset of T). The closure of ψ(T) 1 σ )−1 (W D D is a proper closed subset of T . If t is a closed point in the complement of this closed set, and if t ∈ ϕ−1 (tD ), then σ(t) ∈ W . Therefore σ −1 (W ) is nonempty, and this completes the proof of the theorem. In order to state the next corollary, we introduce some notation. Let f : X → T be a morphism of schemes of ﬁnite type over k, with T integral, and σ : T → X a

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section of f . Suppose that (tm )m≥1 is a dense sequence F of closed points in T such p r that each Xtm is klt around σ(tm ). Suppose that A = j=1 aj j is an R-ideal on X, such that each aj vanishes along σ(T ), but it does not vanish along any ﬁber of f . F For every (not necessarily closed) point ξ ∈ T , we put aj,ξ = aj · OXξ and pj . We also denote by mξ the ideal deﬁning σ(ξ) in Xξ . Aξ = j aj,ξ Corollary B.9. With the above notation and assumptions, the following hold: i) If η is the generic point of T , then Xη is klt at σ(η). Furthermore, there is a subset J of Z>0 such that {ti | i ∈ J} is dense in T , and for every i in J lctσ(η) (Xη , Aη ) = lctσ(ti ) (Xti , Ati ). ii) If E is a divisor over Xη computing lctσ(η) (Xη , Aη ), then after possibly replacing J by a smaller subset J1 with the same properties, we may assume that, in addition, for every i ∈ J1 we have a divisor Ei over Xti that computes lctσ(ti ) (Xti , Ati ), and such that ordE (mη ) = ordEi (mti ) and ordE (aj,η ) = ordEi (aj,ti ) for all j ≤ r. In particular, if E has center equal to σ(η), then each Ei with i ∈ J1 has center σ(ti ). Proof. It is clear that we are allowed to replace T by any open subset V , in which case we need to replace Z>0 by J = {i | ti ∈ V }. Since klt varieties have rational singularities (see [11, Corollary 11.14]), we may apply Theorem B.8 to f . Let U ⊆ T and s be given by this theorem. After replacing T by U , we may assume U = T . Since sKX is Cartier around σ(T ), and since we are only interested in the behavior around σ(T ), we may replace X by the open subset where sKX is Cartier, and therefore assume sKX is Cartier. Consider now a log resolution h : Y → X of (X, A). Let E be the simple normal crossings divisor on Y given by the sum of the h-exceptional divisors and of the divisors appearing in the supports of the ideals aj OY . By generic smoothness, after possibly replacing T by an open subset, we may assume the following properties: (1) The composition f ◦ h is smooth, and E has relative simple normal crossings over T . (2) For every prime divisor Fj in E, its image Zj in X is ﬂat over T and maps onto T . (3) Furthermore, we may and will assume that each Zj contains σ(T ) (otherwise we simply replace T by T 1 σ −1 (Zj )). It follows from (1) that for every (not necessarily closed) point ξ ∈ U , the morphism hξ : Yξ → Xξ is a log resolution of (Xξ , Aξ ). By (2), if F is a component of E that is h-exceptional, then Fξ is hξ -exceptional (note that Fξ is smooth, but might not be connected). Since O(sKX )|Xξ @ O(sKξ ) in a neighborhood of σ(ξ), we deduce that KY /X |Yξ = KYξ /Xξ over the inverse image of an open neighborhood of σ(ξ). Since σ(ξ) ∈ hξ (Fξ ) for every prime divisor F in E, it follows that each Xξ is klt and lctσ(ξ) (Xξ , Aξ ) = lct(X, A). Applying this with ξ = ti and ξ = η gives the assertions in i).

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Tommaso de Fernex, Lawrence Ein, Mircea Mustat¸˘ a

Suppose now that E is as in ii). E appears as a prime divisor on some log resolution of (Xη , Aη · mη ). Since this log resolution is deﬁned over k(η), it follows that after replacing T by a suitable open subset, we may assume that this log resolution is equal to hη for some log resolution h : Y → X as above. In fact, we may assume that h is a log resolution of (X, A · aσ(T ) ), where aσ(T ) is the ideal deﬁning σ(T ) in X. We may again assume that h satisﬁes (1)-(3) above. There is a prime divisor F in E such that E = Fη . It is then clear that, for every i, we may take Ei to be any connected component of Eti , and that the divisors Ei satisfy ii). Remark B.10. It follows from the proof of the corollary that the set J in i) can be chosen independently of the exponents p1 , . . . , pr . In fact, while the convention for R-ideals is that all exponents are positive, it is clear that the result in the corollary still holds if some (but not all) of the pi are allowed to be zero.

References [1] C. Birkar, Ascending chain condition for log canonical thresholds and termination of log ﬂips, Duke Math. J. 136 (2007), 173–180. [2] C. Birkar, P. Cascini, C. Hacon, and J. Mc Kernan, Existence of Minimal Models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468. [3] T. de Fernex, L. Ein, and M. Mustat¸˘ a, Shokurov’s ACC Conjecture for log canonical thresholds on smooth varieties, Duke Math. J. 152 (2010), 93–114. ´ [4] T. de Fernex and M. Mustat¸˘ a, Limits of log canonical thresholds, Ann. Sci. Ecole Norm. Sup´er. (4) 42 (2009), 491–515. [5] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995. [6] R. Elkik, Singularit´es rationnelles et d´eformations, Invent. Math. 47 (1978), 139-147. [7] S.-Y. Jow and E. Miller, Multiplier ideals of sums via cellular resolutions. Math. Res. Lett. 15 (2008), 359–373. [8] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. [9] M. Kawakita, Towards boundedness of minimal log discrepancies by Riemann–Roch theorem, preprint available at arXiv:0903.0418. [10] J. Koll´ ar, Which powers of holomorphic functions are integrable?, preprint available at arXiv:0805.0756. [11] J. Koll´ ar, Singularities of pairs, in Algebraic geometry, Santa Cruz 1995, 221–286, Proc. Symp. Pure Math. 62, Part 1, Amer. Math. Soc., Providence, RI, 1997. [12] J. Koll´ ar and S. Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti, Cambridge Tracts in Mathematics 134, Cambridge University Press, Cambridge, 1998. [13] J. Koll´ ar and S. Mori, Classiﬁcation of three-dimensional ﬂips, J. Amer. Math. Soc. 5 (1992), 533–703.

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[14] R. Lazarsfeld, Positivity in algebraic geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol. 49, Springer-Verlag, Berlin, 2004. [15] H. Matsumura, Commutative ring theory, translated from the Japanese by M. Reid, Second edition, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989. [16] H. Matsumura, Noetherian rings with many derivations, in Contributions to algebra (collection of papers dedicated to Ellis Kolchin), 279–294, Academic Press, New York, 1977. [17] J.-P. Serre, G´eom´etrie alg´ebrique et g´eom´etrie analytique, Ann. Inst. Fourier 6 (1955–1956), 1–42. [18] V. V. Shokurov, Three-dimensional log perestroikas. With an appendix in English by Yujiro Kawamata, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105–203, translation in Russian Acad. Sci. Izv. Math. 40 (1993), 95–202. [19] M. Temkin, Functorial desingularization over Q: boundaries and the embedded case, arXiv: 0912.2570. [20] J.-L. Verdier, Stratiﬁcations de Whitney et th´eor`eme de Bertini–Sard, Invent. Math. 36 (1976), 295–312. Tommaso de Fernex, Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 48112-0090, USA E-mail: [email protected] Lawrence Ein, Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street (M/C 249), Chicago, IL 60607-7045, USA E-mail: [email protected] Mircea Mustat¸a ˘, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA E-mail: [email protected]

Brill-Noether geometry on moduli spaces of spin curves Gavril Farkas∗ The aim of this paper is to initiate a study of geometric divisors of Brill-Noether type on the moduli space S g of spin curves of genus g. The moduli space S g is a compactiﬁcation of the parameter space Sg of pairs [C, η], consisting of a smooth genus g curve C and a theta-characteristic η ∈ Picg−1 (C), see [C]. The study of the birational properties of S g , as well as other moduli spaces of curves with level structure, has received an impetus in recent years, see [BV] [FL], [F2], [Lud], to mention only a few results. Using syzygy divisors, it has been proved in [FL] that the Prym moduli space Rg := Mg (BZ2 ) classifying curves of genus g together with a point of order 2 in the Jacobian variety, is a variety of general type for g ≥ 13 + and g = K 15. The moduli space S g of even spin curves of genus g is known to be of general type for g > 8, uniruled for g < 8, see [F2], whereas the Kodaira dimension + of S 8 is equal to zero, [FV]. This was the ﬁrst example of a naturally deﬁned moduli space of curves of genus g ≥ 2, having intermediate Kodaira dimension. An application of the main construction presented in this paper, gives a new way of + computing the class of the divisor Θnull of vanishing theta-nulls on S g , reproving thus the main result of [F2]. Virtually all attempts to show that a given moduli space Mg,n is of general type, rely on the calculation of the class of a certain eﬀective divisor D ⊂ Mg,n enjoying extremality properties in its respective eﬀective cone of divisors Eﬀ(Mg,n ), so that the canonical class KMg,n lies in the cone spanned by [D], boundary classes δi:S , tautological classes λ, ψ1 , . . . , ψn , and possibly other eﬀective geometric classes. Examples of such a program being carried out can be found in [EH2], [HM]-for the case of Brill-Noether divisors on Mg consisting of curves with a grd when ρ(g, r, d) = −1, [Log]-where pointed Brill-Noether divisors on Mg,n are studied, and [F1]-for the case of Koszul divisors on Mg , which provide counterexamples to the Slope Conjecture on Mg . A natural question is what the analogous geometric divisors on the spin moduli space of curves S g should be? In this paper we propose a construction for spin Brill-Noether divisors on both + − spaces S g and S g , deﬁned in terms of the relative position of theta-characteristics with respect to diﬀerence varieties on Jacobians. Precisely, we ﬁx integers r, s ≥ 1 such that d := rs + r ≡ 0 mod 2, and then set g := rs + s. One can write d = 2i. By standard Brill-Noether theory, a general curve [C] ∈ Mg carries a ﬁnite number of (necessarily complete and base point free) linear series grd . One considers the following loci of spin curves (both odd and even) r Ug,d := {[C, η] ∈ Sg∓ : ∃L ∈ Wdr (C) such that η ⊗ L∨ ∈ Cg−i−1 − Ci }. ∗ Research

partially supported by Sonderforschungsbereich 647 ”Raum-Zeit-Materie”.

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G. Farkas |η⊗L|

r Thus Ug,d consists of spin curves such that the embedded curve C −→ Pd−1 r admits an i-secant (i − 2)-plane. We shall prove that for s ≥ 2, the locus Ug,d is ∓ always a divisor on Sg , and we ﬁnd a formula for the class of its compactiﬁcation ∓

in S g . For simplicity, we display this formula in the introduction only in the case r = 1, when g ≡ 2 mod 4: Theorem 0.1. We ﬁx an integer a ≥ 1 and set g := 4a + 2. The locus ∓ 1 1 U4a+2,2a+2 := {[C, η] ∈ S4a+2 : ∃L ∈ W2a+2 (C) such that η ⊗ L∨ ∈ C3a − Ca+1 } ∓

is an eﬀective divisor and the class of its compactiﬁcation in S g is given by the formula O $ O$ Q7 G a+2 4a 4a + 2 1 U 4a+2,2a+2 ≡ 192a3 + 736a2 + 692a + 184 λ 8(2a + 1)(4a + 1) 2a a 3 7 G 7 G ∓ − 32a3 + 104a2 + 82a + 19 α0 − 64a3 + 176a2 + 148a + 36 β0 − · · · ∈ Pic(S g ). To specialize further, in Theorem 0.1 we set a = 1, and ﬁnd the class of (the closure of) the locus of spin curves [C, η] ∈ S6∓ , such that there exists a pencil |η⊗L|

L ∈ W41 (C) for which the linear series C −→ P3 is not very ample: 1

U 6,4 ≡ 451λ −

237 ∓G α0 − 106β0 − · · · ∈ Pic(S 6 . 4

g−1 The case s = 1, when necessarily L = KC ∈ W2g−2 (C), produces a divisor only +

on S g , and we recover in this way the main calculation from [F2], used to prove +

that S g is a variety of general type for g > 8. We recall that Θnull := {[C, η] ∈ Sg+ : H 0 (C, η) K= 0} denotes the irreducible divisor of vanishing theta-nulls. +

Theorem 0.2. Let π : S g → Mg be the ramiﬁed covering which forgets the spin g−1

structure. For g ≥ 3, one has the equality U g,2g−2 = 2 · Θnull of codimension 1+

cycles on the open subvariety π −1 (Mg ∪ Δ0 ) of S g . Moreover, there is an equality of classes g−1

U g,2g−2 ≡ 2 · Θnull ≡

1 1 + λ − α0 − 0 · β0 − · · · ∈ Pic(S g ). 2 8

We point out once more the low slope of the divisor Θnull . No similar divisor with such remarkable class is known to exist on Rg . In Section 4, we present a third way of calculating the class [Θnull ], by rephrasing the condition that a curve C have a vanishing theta-null η, if and only if, for a pencil A on C of minimal degree, the multiplication map of sections H 0 (C, A) ⊗ H 0 (C, A ⊗ η) → H 0 (C, A⊗2 ⊗ η)

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is not an isomorphism. In this way, Θnull appears as the push-forward of a degeneracy locus of a morphism between vector bundles of the same rank deﬁned over a Hurwitz stack of coverings. In the last section of the paper, we study the divisor Θg,1 on the universal curve Mg,1 , which consists of points in the support of odd theta-characteristics. This divisor, somewhat similar to the divisor W g of Weierstrass points on Mg,1 , cf. [Cu], should be of some importance in the study of the birational geometry of Mg,1 : Theorem 0.3. The class of the compactiﬁcation in Mg,1 of the eﬀective divisor Θg,1 := {[C, q] ∈ Mg,1 : q ∈ supp(η) for some [C, η] ∈ Sg− } is given by the following formula in Pic(Mg,1 ): g−1 Q 3 M 7 G (2i + 1)(2g−i − 1)δi . Θg,1 ≡ 2g−3 (2g − 1) λ + 2ψ − 2g−3 δirr − (2g − 2)δ1 − i=1

When g = 2, the divisor Θ2 specializes to the divisor of Weierstrass points: Θ2,1 = W2 := {[C, q] ∈ M2,1 : q ∈ C is a Weierstrass point}. If we use Mumford’s formula λ = δ0 /10 + δ1 /5 ∈ Pic(M2 ), Theorem 0.3 reads Θ2,1 ≡

1 3 3 λ + 3ψ − δirr − δ1 = −λ + 3ψ − δ1 ∈ Pic(M2,1 ), 2 4 2

that is, we recover the formula for the class of the Weierstrass divisor on M2,1 , cf. [EH2]. When g = 3, the condition [C, q] ∈ Θ3,1 , states that the point q ∈ C lies on |KC |

one of the 28 bitangent lines of the canonically embedded curve C −→ P2 . Corollary 0.4. The class of the compactiﬁcation in M3,1 of the bitangent locus Θ3,1 := {[C, q] ∈ M3,1 : q lies on a bitangent of C} is equal to Θ3,1 ≡ 7λ + 14ψ − δirr − 9δ1 − 5δ2 ∈ Pic(M3,1 ). If p : Mg,1 → Mg is the map forgetting the marked point, we note the equality 1

D3 ≡ p∗ (M3,2 ) + 2 · W 3 + 2ψ ∈ Pic(M3,1 ), where W 3 ≡ −λ + 6ψ − 3δ1 − δ2 is the divisor of Weierstrass points on M3,1 and 1 M3,2 is the hyperelliptic locus in M3 . Since the class ψ ∈ Pic(M3,1 ) is big and nef, it follows that Θ3,1 (unlike the divisor Θ2,1 ∈ Pic(M2,1 )), lies in the interior of the cone of eﬀective divisors Eﬀ(M3,1 ), or in other words, it is big. In particular, it cannot be contracted by a rational map M3,1 ''( X to any projective variety X. This phenomenon extends to all higher genera: Corollary 0.5. For every g ≥ 3, the divisor Θg,1 ∈ Eﬀ(Mg,1 ) is big. It is not known whether the Weierstrass divisor W g lies on the boundary of the eﬀective cone Eﬀ(Mg,1 ) for g suﬃciently large.

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1. Generalities about S g As usual, we follow the convention that if M is a Deligne-Mumford stack, then M denotes its associated coarse moduli space. We ﬁrst recall basic facts about Cornalba’s stack of stable spin curves π : Sg → Mg , see [C], [F2], [Lud] for details and other basic properties. If X is a nodal curve, a smooth rational component R ⊂ X is said to be exceptional if #(R ∩ X − R) = 2. The curve X is said to be quasi-stable if #(R ∩ X − R) ≥ 2 for any smooth rational component R ⊂ X, and moreover, any two exceptional components of X are disjoint. A quasi-stable curve is obtained from a stable curve by possibly inserting a rational curve at each of its nodes. We denote by [st(X)] ∈ Mg the stable model of the quasi-stable curve X. Deﬁnition 1.1. A spin curve of genus g consists of a triple (X, η, β), where X is a genus g quasi-stable curve, η ∈ Picg−1 (X) is a line bundle of degree g − 1 such that ηR = OR (1) for every exceptional component R ⊂ X, and β : η ⊗2 → ωX is a sheaf homomorphism which is generically non-zero along each non-exceptional component of X. Stable spin curves of genus g form a smooth Deligne-Mumford stack Sg which + − splits into two connected components Sg and Sg , according to the parity of h0 (X, η). Let f : C → Sg be the universal family of spin curves of genus g. In particular, for every point [X, η, β] ∈ S g , there is an isomorphism between f −1 ([X, η, β]) and the quasi-stable curve X. There exists a (universal) spin line bundle P ∈ Pic(C) of relative degree g − 1, as well as a morphism of OC -modules B : P ⊗2 → ωf having the property that P|f −1 ([X,η,β]) = η and B|f −1 ([X,η,β]) = β : η ⊗2 → ωX , for all spin curves [X, η, β] ∈ S g . Throughout we use the canonical isomorphism Pic(Sg )Q ∼ = Pic(S g )Q and we make little distinction between line bundles on the stack and the corresponding moduli space. 1.1. The boundary divisors of S g . We discuss the structure of the boundary divisors of S g and concentrate on + − the case of S g , the diﬀerences compared to the situation on S g being minor. We describe the pull-backs of the boundary divisors Δi ⊂ Mg under the map π. First we ﬁx an integer 1 ≤ i ≤ [g/2] and let [X, η, β] ∈ π −1 ([C ∪y D]), where [C, y] ∈ Mi,1 and [D, y] ∈ Mg−i,1 . For degree reasons, then X = C ∪y1 R ∪y2 D, where R is an exceptional component such D ∩ R = {y2 }. 7 G that C ∩ R = {y1 } and ⊗2 Furthermore η = ηC , ηD , ηR = OR (1) ∈ Picg−1 (X), where ηC = KC and ⊗2 ηD = KD . The theta-characteristics ηC and ηD have the same parity in the case + − + of S g (and opposite parities for S g ). One denotes by Ai ⊂ S g the closure of the locus corresponding to pairs of pointed spin curves 7 G + + [C, y, ηC ], [D, y, ηD ] ∈ Si,1 × Sg−i,1 +

and by Bi ⊂ S g the closure of the locus corresponding to pairs 7 G − − [C, y, ηC ], [D, y, ηD ] ∈ Si,1 × Sg−i,1 .

Brill-Noether geometry on moduli spaces of spin curves

263

+

If α := [Ai ], βi := [Bi ] ∈ Pic(Sg ), we have the relation π ∗ (δi ) = αi + βi . Next, we describe π ∗ (δ0 ) and pick a stable spin curve [X, η, β] such that st(X) = Cyq := C/y ∼ q, with [C, y, q] ∈ Mg−1,2 . There are two possibilities depending on whether X possesses an exceptional component or not. If X = Cyq and ηC := ν ∗ (η) ⊗2 where ν : C → X denotes the normalization map, then ηC = KC (y + q). For each g−1 choice of ηC ∈ Pic (C) as above, there is precisely one choice of gluing the ﬁbres ηC (y) and ηC (q) such that h0 (X, η) ≡ 0 mod 2. We denote by A0 the closure in + ⊗2 = KC (y + q)] as above. S g of the locus of points [Cyq , ηC ∈ Picg−1 (C), ηC If X = C ∪{y,q} R, where R is an exceptional component, then ηC := η ⊗ OC is a theta-characteristic on C. Since H 0 (X, ω) ∼ = H 0 (C, ωC ), it follows that [C, ηC ] ∈ + + Sg−1 . We denote by B0 ⊂ S g the closure of the locus of points B H 9 + C ∪{y,q} R, ηC ∈ KC , ηR = OR (1) ∈ S g . A local analysis carried out in [C], shows that B0 is the branch locus of π and the + + ramiﬁcation is simple. If α0 = [A0 ] ∈ Pic(S g ) and β0 = [B0 ] ∈ Pic(S g ), we have the relation π ∗ (δ0 ) = α0 + 2β0 . (1)

2. Diﬀerence varieties and theta-characteristics We describe a way of calculating the class of a series of eﬀective divisors on both − + moduli spaces S g and S g , deﬁned in terms of the relative position of a thetacharacteristic with respect to the divisorial diﬀerence varieties in the Jacobian of a curve. These loci, which should be thought of as divisors of Brill-Noether type on S g , inherit a determinantal description over the entire moduli stack of spin curves, via the interpretation of diﬀerence varieties in Picg−2i−1 (C) as Raynaud theta-divisors for exterior powers of Lazarsfeld bundles provided in [FMP]. The ) g of determinantal description is then extended over a partial compactiﬁcation S Sg , using the explicit description of stable spin curves. The formulas we obtain − + for the class of these divisors are identical over both Sg and Sg , therefore we ∓

sometimes use the symbol Sg (or even Sg ), to denote one of the two spin moduli spaces. ∨ We start with a curve [C] ∈ Mg and denote as usual by QC := MK the C associated Lazarsfeld bundle [L] deﬁned via the exact sequence on C ev

0 → MKC → H 0 (C, KC ) ⊗ OC → KC → 0. Note that QC is a semistable vector bundle on C (even stable, when the curve C is non-hyperelliptic), and µ(QC ) = 2. For integers 0 ≤ i ≤ g − 1, one deﬁnes the divisorial diﬀerence variety Cg−i−1 − Ci ⊂ Picg−2i−1 (C) as being the image of the diﬀerence map φ : Cg−i−1 × Ci → Picg−2i−1 (C),

φ(D, E) := OC (D − E).

264

G. Farkas

When C is non-hyperelliptic, φ is a birational isomorphism. The main result from [FMP] provides a scheme-theoretic identiﬁcation of divisors on the Jacobian variety Cg−i−1 − Ci = Θ∧i QC ⊂ Picg−2i−1 (C),

(2)

where the right-hand-side denotes the Raynaud locus [R] Θ∧i QC := {η ∈ Picg−2i−1 (C) : H 0 (C, ∧i QC ⊗ η) K= 0}. The non-vanishing H 0 (C, ∧i QC ⊗ ξ) K= 0 for all line bundles ξ = OC (D − E), where D ∈ Cg−i−1 and E ∈ Ci , follows from [L]. The thrust of [FMP] is that the reverse inclusion Θ∧i QC ⊂ Cg−i−1 −Ci also holds. Moreover, identiﬁcation (2) shows that, somewhat similarly to Riemann’s Singularity Theorem, the product Cg−i−1 × Ci can be thought of as a canonical desingularization of the generalized theta-divisor Θ∧i QC . We ﬁx integers r, s > 0 and set d := rs + r and g := rs + s, therefore the Brill-Noether number ρ(g, r, d) = 0. We assume moreover that d ≡ 0 mod 2, that is, either r is even or s is odd, and write d = 2i. We deﬁne the following locus in the spin moduli space Sg∓ : r Ug,d := {[C, η] ∈ Sg∓ : ∃L ∈ Wdr (C) such that η ⊗ L∨ ∈ Cg−i−1 − Ci }. r Using (2), the condition [C, η] ∈ Ug,d can be rewritten in a determinantal way as

H 0 (C, ∧i MKC ⊗ η ⊗ L) K= 0. Tensoring by η ⊗ L the exact sequence coming from the deﬁnition of MKC , namely 0 −→ ∧i MKC −→ ∧i H 0 (C, KC ) ⊗ OC −→ ∧i−1 MKC ⊗ KC −→ 0, then taking global sections and ﬁnally using that MKC (hence all of its exterior r powers) are semi-stable vector bundles, we ﬁnd that [C, η] ∈ Ug,d if and only if the map 7 G φ(C, η, L) : ∧i H 0 (C, KC ) ⊗ H 0 C, η ⊗ L) → H 0 (C, ∧i−1 MKC ⊗ KC ⊗ η ⊗ L (3) is not an isomorphism for a certain L ∈ Wdr (C). Since µ(∧i−1 MKC ⊗KC ⊗η ⊗L) ≥ 2g − 1 and ∧i−1 MKC is a semi-stable vector bundle on C, it follows that $ O g d. h0 (C, ∧i−1 MKC ⊗ KC ⊗ η ⊗ L) = χ(C, ∧i−1 MKC ⊗ KC ⊗ η ⊗ L) = i We assume that h1 (C, η⊗L) = 0. This condition is satisﬁed outside a locus of Sg∓ of codimension at least 2; in fact, if H 1 (C, η ⊗ L) K= 0, then H 1 (C, KC ⊗ L⊗(−2) ) = K 0, in particular the Petri map µ0 (C, L) : H 0 (C, L) ⊗ H 0 (C, KC ⊗ L∨ ) → H 0 (C, KC )

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is not injective. Then h0 (C, L ⊗ η) = d and we note that φ(C, η, L) is a map between vector spaces of the same rank. This obviously suggests a determinantal r presentation of Ug,d as the (push-forward of) a degeneracy locus between vector bundles of the same rank. In what follows we extend this presentation over a ∓ partial compactiﬁcation of Sg . We refer to [FL] Section 2 for a similar calculation over the Prym moduli stack Rg . We denote by M0g ⊂ Mg the open substack classifying curves [C] ∈ Mg such r that Wd−1 (C) = ∅, Wdr+1 (C) = ∅ and moreover H 1 (C, L ⊗ η) = 0, for every r L ∈ Wd (C) and each odd-theta characteristic η ∈ Picg−1 (C). From general Brill)0 ⊂ Noether theory one knows that codim(Mg − M0g , Mg ) ≥ 2. Then we deﬁne Δ Δ0 to be the open substack consisting of 1-nodal stable curves [Cyq := C/y ∼ q], where [C] ∈ Mg−1 is a curve satisfying the Brill-Noether theorem and y, q ∈ C. We set 0 ) 0, Mg := M0g ∪ Δ 0

0

0

0

and then Sg := π −1 (Mg ) = (Sg )+ ∪ (Sg )− . Following [EH1], [F1], we consider the proper Deligne-Mumford stack 0

σ0 : Grd → Mg 0

classifying pairs [C, L] with [C] ∈ Mg and L ∈ Wdr (C). For any curve [C] ∈ 0

Mg and L ∈ Wdr (C), we have that h0 (C, L) = r + 1, that is, Grd parameterizes ) 0 , we have the only complete linear series. For a point [Cyq := C/y ∼ q] ∈ Δ identiﬁcation 9 H σ0−1 Cyq = {L ∈ Wdr (C) : h0 (C, L ⊗ OC (−y − q)) = r}, that is, we view linear series on singular curves as linear series on the normalization such that the divisor of the nodes imposes only one condition. We denote by 0

fdr : Crg,d := Mg,1 ×M0 Grd → Grd g

0

0

the pull-back of the universal curve p : Mg,1 → Mg to Grd . Once a Poincar´e bundle L on Crg,d has been chosen, we form the three codimension 1 tautological classes in A1 (Grd ): G 7 G 7 7 7 G G a := (fdr )∗ c1 (L)2 , b := (fdr )∗ c1 (L) · c1 (ωfdr ) , c := (fdr )∗ c1 (ωfdr )2 = (σ0 )∗ κ1 . (4) The dependence of a, b, c on the choice of L is discussed in both [F2] and [FL]. We introduce the stack of grd ’s on spin curves 0

0

0

0

σ : Grd (Sg /Mg ) := Sg ×M0 Grd → Sg g

and then the corresponding universal spin curve over the grd parameter space 0

0

0

0

f D : Cdr := C ×S0 Grd (Sg /Mg ) → Grd (Sg /Mg ). g

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We note that f D is a family of quasi-stable curves carrying at the same time a spin 0 0 structure as well as a grd . Just like in [FL], the boundary divisors of Grd (Sg /Mg ) are denoted by the same symbols, that is, one sets AD0 := σ ∗ (AD0 ) and B0D := σ ∗ (B0D ) and then 0 0 α0 := [AD0 ], β0 := [B0D ] ∈ A1 (Grd (Sg /Mg )). We observe that two tautological line bundles live on Cdr , namely the pull-back of the universal spin bundle Pdr ∈ Pic(Cdr ) and a Poincar´e bundle L ∈ Pic(Cdr ) singling out the grd ’s, that is, L|f "−1 [X,η,β,L] = L ∈ Wdr (C), for each point [X, η, β, L] ∈ 0

0

0

0

Grd (Sg /Mg ). Naturally, one also has the classes a, b, c ∈ A1 (Grd (Sg /Mg )) deﬁned by the formulas (4). The following result is easy to prove and we skip details: 0

0

Proposition 2.1. We denote by f D : Cdr → Grd (Sg /Mg ) the universal quasi-stable spin curve and by Pdr ∈ Pic(Cdr ) the universal spin bundle of relative degree g − 1. 0 0 One has the following formulas in A1 (Grd (Sg /Mg )): (1) f∗D (c1 (ωf " ) · c1 (Pdr )) = 21 c. (2) f∗D (c1 (Pdr )2 ) = 41 c − 21 β0 . (3) f∗D (c1 (L) · c1 (Pdr )) = 21 b. r We determine the class of a compactiﬁcation of Ug,d by pushing forward a 0

0

0

codimension 1 degeneracy locus via the map σ : Grd (Sg /Mg ) → Sg . To that end, 0

0

we deﬁne a sequence of tautological vector bundles on Grd (Sg /Mg ): First, for l ≥ 0 we set A0,l := f∗D (L ⊗ ωf⊗l" ⊗ Pdr ). It is easy to verify that R1 f∗D (L ⊗ ωf⊗l" ⊗ Pdr ) = 0, hence A0,l is locally free over 0

0

⊗l Grd (Sg /Mg ) of rank equal to h0 (X, L ⊗ ωX ⊗ η) = l(2g − 2) + d. Next we introduce the global Lazarsfeld vector bundle M over Cdr by the exact sequence 7 G 0 −→ M −→ (f D )∗ f∗D ωf " −→ ωf " −→ 0, 0

0

and then for all integers a, j ≥ 1 we deﬁne the sheaf over Grd (Sg /Mg ) r Aa,j := f∗D (∧a M ⊗ ωf⊗j " ⊗ L ⊗ Pd ).

In a way similar to [FL] Proposition 2.5, one shows that 7 G ⊗(i−a) R1 f∗D ∧a M ⊗ ωf " ⊗ L ⊗ Pdr = 0, 0

0

therefore by Grauert’s theorem Aa,i−a is a vector bundle over Grd (Sg /Mg ) of rank $ O 7 G g−1 ⊗(i−a) . rk(Aa,i−a ) = χ X, ∧a MωX ⊗ ωX ⊗ L ⊗ η = 2(i − a)g a

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Furthermore, for all 1 ≤ a ≤ i − 1, the vector bundles Aa,i−a sit in exact sequences 0 −→ Aa,i−a −→ ∧a f∗D (ωf " ) ⊗ A0,i−a −→ Aa−1,i−a+1 −→ 0,

(5) ⊗(i−a)

where the right exactness boils down to showing that H 1 (X, ∧a MωX ⊗ ωX 0 0 η ⊗ L) = 0 for all [X, η, β, L] ∈ Grd (Sg /Mg ). 0

⊗

0

We denote as usual E := f∗D (ωf " ) the Hodge bundle over Grd (Sg /Mg ) and then note that there exists a vector bundle map φ : ∧i E ⊗ A0,0 → Ai−1,1 0

(6)

0

between bundles of the same rank over Grd (Sg /Mg ). For [C, η, L] ∈ σ −1 (M0g ) the ﬁbre of this morphism is precisely the map φ(C, η, L) deﬁned by (3). Theorem 2.2. The vector bundle morphism φ : ∧i E⊗A0,0 → Ai−1,1 is generically 0 0 r non-degenerate over Grd (Sg /Mg ). It follows that Ug,d is an eﬀective divisor over + − Sg for all s ≥ 1, and over Sg as well for s ≥ 2. Proof. We specialize C to a hyperelliptic curve, and denote by A ∈ W21 (C) the hyperelliptic involution. The Lazarsfeld bundle splits into a sum of line bundles QC ∼ = A⊕(g−1) , therefore the condition H 0 (C, ∧i MKC ⊗ η ⊗ L) = 0 translates into H 0 (C, η ⊗ A⊗i ⊗ L∨ ) = 0. Suppose that h0 (C, η ⊗ A⊗i ⊗ L∨ ) ≥ 1 for any L = A⊗r ⊗ OC (x1 + · · · + xd−2r ) ∈ Wdr (C), where the x1 , . . . , xd−2r ∈ C are arbitrarily chosen points. This implies that h0 (C, η ⊗ A⊗(i−r) ) ≥ d − 2r + 1. Any theta-characteristic on C is of the form η = A⊗m ⊗ OC (p1 + · · · + pg−2m−1 ), where 1 ≤ m ≤ (g − 1)/2 and p1 , . . . , pg−2m−1 ∈ C are Weierstrass points. Choosing a theta-characteristic on C for which m ≤ i − r − 1 (which can be done in all cases except on Sg− when i = r), we obtain that h0 (C, η ⊗ A⊗(i−r) ) ≤ d − 2r, a contradiction. Proof of Theorem 0.1. To compute the class of the degeneracy locus of φ we use repeatedly the exact sequence (5). We write the following identities in the 0 0 codimension 1 Chow group A1 (Grd (Sg /Mg )): i 7 G M c1 Ai−1,1 − ∧i E ⊗ A0,0 = (−1)l−1 c1 (∧i−l E ⊗ A0,l ) = l=0

=

i M l=0

$ O $ O 3 Q g−1 g c1 (A0,l ) . c1 (E) + (−1)l+1 (2l(g − 1) + d) i−l−1 i−l

Using Proposition 2.1 one can show via the Grothendieck-Riemann-Roch formula 0 0 applied to f D : Cdr → Grd (Sg /Mg ) that one has that c1 (A0,l ) = λ +

Q l2 2

−

1 13 1 0 0 c + a + lb − β0 ∈ A1 (Grd (Sg /Mg )). 8 2 4

268

G. Farkas

7 G To determine σ∗ c1 (Ai−1,1 − ∧i E) ∈ A1 (Sg ) we use [F1], [Kh]: If N := deg(σ) = #(Wdr (C)) denotes the number of grd ’s on a general curve [C] ∈ Mg , then there exists a precisely described choice of a Poincar´e bundle on Crg,d such that the push-forwards 0

0

of the tautological classes on Grd (Sg /Mg ) are given as follows (cf. [F1], [Kh] and especially [FL] Section 2, for a similar argument in the Prym case): 3 Q dN 1 σ∗ (a) = (gd−2g 2 +8d−8g+4)λ+ (2g 2 −gd+3g−4d−2)(α0 +2β0 ) (g − 1)(g − 2) 6 and σ∗ (b) =

3 dN Q 0 0 12λ − α0 − 2β0 ∈ A1 (Grd (Sg /Mg )). 2g − 2 0

0

One notes that c1 (Ai−1,1 − ∧i E ⊗ A0,0 ) ∈ A1 (Grd (Sg /Mg )) does not depend on the Poincar´e bundle. Using the previous formulas, after some arithmetic, one computes r and ﬁnishes the proof. ✷ the class of the partial compactiﬁcation of Ug,d When s = 2a + 1, hence g = (2a + 1)(r + 1) and d = 2r(a + 1), our calculation shows that 7 r ¯ λ − α¯0 α0 − β¯0 β0 − · · · ) ∈ Pic(S ∓ ), U g,d ≡ ca,r λ g where ca,r ∈ Q>0 is explicitly known and ¯ = 12r3 − 12r2 − 48a2 + 96a3 + 48r4 a + 2208r3 a3 + 1968r3 a2 + 3936r2 a3 + 2208ra3 λ + 552r3 a + 3984r2 a2 + 1080r2 a + 2160ra2 + 528ra + 192r4 a4 + 384r4 a3 + 768r3 a4 + 960r2 a4 + 240r4 a2 + 384ra4 , α ¯ 0 = 220ra2 + 536r2 a3 + 32r4 a4 + 36ra + 24a3 + 328r3 a3 + 296ra3 + 8r4 a + 64r4 a3 + 3r3 + 468r2 a2 + 128r3 a4 + 74r3 a + 40r4 a2 + 160r2 a4 + 64ra4 + 268r3 a2 + 110r2 a − 3r2 − 12a2 ,

and β¯0 = 96ra + 64r4 a4 + 16r4 a + 416ra2 + 928r2 a3 + 448ra3 + 208r2 a + 608r3 a3 + 256r3 a4 + 112r3 a + 80r4 a2 + 320r2 a4 + 128ra4 + 464r3 a2 + 128r4 a3 + 816r2 a2 .

These formulas, though unwieldy, carry a great deal of information about S g . In the simplest case, s = 1 (that is, a = 0) and r = g − 1, then necessarily L = g−1 (C) and the condition η − KC ∈ −Cg−1 is equivalent to H 0 (C, η) K= 0. KC ∈ W2g−2 +

In this way we recover the theta-null divisor Θnull on S g , or more precisely also taking into account multiplicities [F2], g−1 Ug,2g−2 = 2 · Θnull .

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At the same time, on Sg+ one does not get a divisor at all. In particular, we ﬁnd that 1 1 g−1 + U g,2g−2 ≡ 2 · Θnull ≡ λ − α0 − 0 · β0 − · · · ∈ Pic(S g ). 2 8 2 Another interesting case is when r = 2, hence g = 3s, L ∈ W2s+2 (C) and the ∨ condition η ⊗ L ∈ C2s−2 − Cs+1 is equivalent to requiring that the embedded |η⊗L|

curve C −→ P2s+1 has an (s + 1)-secant (s − 1)-plane: ∓

Theorem 2.3. For g = 3s, d = 2s+2, the class of the closure in Sg of the eﬀective divisor ∓ 2 2 Ug,d := {[C, η] ∈ S3s : ∃L ∈ W2s+2 (C) such that η ⊗ L∨ ∈ C2s−2 − Cs+1 } ∓

is given by the formula in Pic(S g ): 2 U g,d

≡

! ! “ ´ g g 1 4(216s4 +513s3 −348s2 −387s+18 λ 2 2 s+2 s, s, s 24g(g − 1) (g − 2)(s + 1)

” ` ´ ` ´ − 144s4 + 225s3 − 268s2 − 99s + 10 α0 − 288s4 + 288s3 + 320s2 + 32 β0 − · · · .

For instance, for g = 9, we obtain the class of the closure of the locus of those spin curves [C, η] ∈ S9∓ , for which there exists a net L ∈ W82 (C) such that η ⊗ L∨ ∈ C4 − C4 : 2

U 9,8 ≡ 235 · 35

Q 36 5

λ − α0 −

3 428 ∓ β0 − · · · ∈ Pic(S 9 ). 235

+

3. The class of Θnull on S g : An alternative proof using the Hurwitz stack We present an alternative way of computing the class of the divisor [Θnull ] (in even genus), as the push-forward of a determinantal cycle on a Hurwitz scheme of degree k coverings of genus g curves. We set g = 2k − 2, r = 1, d = k, hence ρ(g, 1, k) = 0, and use the notation from the previous section. In particular, 0 we have the proper morphism σ0 : G1k → Mg from the Hurwitz stack of g1k ’s, and the universal spin curve over the Hurwitz stack 0

0

0

0

f D : C1k := C ×S0 G1k (Sg /Mg ) → G1k (Sg /Mg ). g

0

0

Once more, we introduce a number of vector bundles over G1k (Sg /Mg ): First, we set H := f∗D (L). By Grauert’s theorem, H is a vector bundle of rank 2 over

270

G. Farkas 0

0

G1k (Sg /Mg ), having ﬁbre H[X, η, β, L] = H 0 (X, L), where L ∈ Wk1 (X). Then for j ≥ 1 we deﬁne Bj := f∗D (L⊗j ⊗ Pk1 ). 0

0

Since R1 f∗D (L⊗j ⊗ Pk1 ) = 0, we ﬁnd that Bj is a vector bundle over G1k (Sg /Mg ) of rank equal to h0 (X, L⊗j ⊗ η) = kj. 0

0

Proposition 3.1. If a, b, c are the tautological classes on G1k (Sg /Mg ) in codimension 1 deﬁned by (4), then for all j ≥ 1 one has the following formula in 0 0 A1 (G1k (Sg /Mg )): j2 j 1 1 c1 (Bj ) = λ − c + a − b − β0 . 8 2 2 4 Proof. We apply the Grothendieck-Riemann-Roch formula to the universal curve 0 0 over the Hurwitz stack f D : Ck1 → G1k (Sg /Mg ) and write: G 7 c1 (Bj ) = c1 f!D (L⊗j ⊗ Pk1 ) = = f∗D

5Q

1 + c1 (L⊗j ⊗ Pk1 ) +

c21 (L⊗j ⊗ Pk1 ) 3Q c1 (ωf " ) c21 (ωf " ) + [Sing(f D )] 3S 1− + , 2 2 12 2

where Sing(f D ) ⊂ Xk1 denotes the codimension 2 singular locus of the morphism f D , therefore f∗D [Sing(f D )] = α0 + 2β0 . We then use Mumford’s formula [HM] pulled 0 0 0 back from Mg to G1k (Sg /Mg ), to write that κ1 = f∗D (c21 (ωf " )) = 12λ − (α0 + 2β0 ) and then note that f∗D (c1 (L) · c1 (Pk1 )) = 0 (the restriction of L to the exceptional 0 0 divisor of f D : Ck1 → G1k (Sg /Mg ) is trivial). Similarly, we note that f∗D (c1 (ωf " ) · c1 (Pk1 )) = c/2. Finally, we write that f∗D (c21 (Pk1 )) = c/4 − β0 /2. 0

0

For j ≥ 1 there are natural vector bundle morphisms over G1k (Sg /Mg ) χj : H ⊗ Bj → Bj+1 . Over a point [C, ηC , L] ∈ Sg+ ×Mg G1k corresponding to an even theta-characteristic ηC and a pencil L ∈ Wk1 (C), the morphism χj is given by multiplications of global sections χj [C, η, L] : H 0 (C, L) ⊗ H 0 (C, L⊗j ⊗ ηC ) → H 0 (C, L⊗(j+1) ⊗ ηC ). In particular, χ1 : H ⊗ B1 → B2 is a morphism between vector bundles of the same rank. From the base point free pencil trick, the degeneration locus Z1 (χ1 ) is G 7 0 (set-theoretically) equal to the inverse image σ −1 Θnull ∩ (S g )+ .

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Brill-Noether geometry on moduli spaces of spin curves

Theorem 3.2. We ﬁx g = 2k − 2. The vector bundle morphism χ1 : H ⊗ B1 → B2 0 0 deﬁned over G1k (Sg /Mg ) is generically non-degenerate and we have the following formula for the class of its degeneracy locus: [Z1 (χ1 )] = c1 (B2 − H ⊗ B1 ) =

1 1 0 0 λ − α0 + a − kc1 (H) ∈ A1 (G1k (Sg /Mg )). 2 8 +

The class of the push-forward σ∗ [Z1 (χ1 )] in Pic(S g ) is given by the formula: 3 2(2k − 2)! 7 G (2k − 2)! Q 1 1 σ∗ c1 (B2 − H ⊗ B1 ) ≡ Θ λ − α0 − 0 · β0 ≡ k!(k − 1)! 2 8 k!(k − 1)! null

+

|S g

.

Proof. The ﬁrst part follows directly from Theorem 3.1. To determine the push0 forward of codimension 1 tautological classes to (S g )+ , we use again [F1], [Kh]: 0

0

One writes the following relations in A1 ((Sg )+ ) = A1 ((S g )+ ): Q 3k(k + 1) 3 k2 0 σ∗ (a) = deg(G1k /Mg ) − λ+ (α0 + 2β0 ) , 2k − 3 2(2k − 3) 0

σ∗ (b) = deg(G1k /Mg )

Q 6k 3 k λ− (α0 + 2β0 ) , 2k − 3 2(2k − 3)

and Q 3 7 k k+1 0 λ+ (α0 + 2β0 ) , σ∗ c1 (H)) = deg(G1k /Mg ) −3 2k − 3 2(2k − 3) where

0

N := deg(G1k /Mg ) =

(2k − 2)! k!(k − 1)!

denotes the Catalan number of linear series g1k on a general curve of genus 2k − 2. This yields yet another proof of the main result from [F2], in the sense that we compute the class of the divisor Θnull of vanishing theta-nulls: 1 G 0 71 0 σ∗ (Z1 (χ1 )) = deg(G1k /Mg ) λ − α0 ≡ 2deg(G1k /Mg ) [Θnull 2 8

0

|(S g )+

].

Remark 3.3. The multiplicity 2 appearing in the expression of σ∗ (Z1 (χ1 )) is justiﬁed by the fact that dim Ker(χ1 (t)) = h0 (C, ηC ) for every [C, ηC , L] ∈ σ −1 ((Sg0 )+ ). This of course is always an even number. Thus we have the equality of cycles 0

0

Z1 (χ1 ) = Z2 (χ1 ) = {t ∈ Grd (Sg /Mg ) : co-rank(φ1 (t)) ≥ 2}, that is, χ1 degenerates in codimension 1 with corank 2, and Z1 (χ1 ) is an everywhere non-reduced scheme.

272

G. Farkas

4. The divisor of points of odd theta-characteristics In this section we compute the class of the divisor Θg,1 . The study of geometric divisors on Mg,1 begins with [Cu], where the locus of Weierstrass points is determined: $ Wg ≡ −λ +

O O g−1 $ M g−i+1 g+1 δi:1 ∈ Pic(Mg,1 ). ψ− 2 2 i=1

More generally, if α ¯ : 0 ≤ α0 ≤ . . . ≤ αr ≤ d − r is a Schubert index of type (r, d) with r M αi = −1, ρ(g, r, d, α ¯ ) := ρ(g, r, d) − i=0

one deﬁnes the pointed Brill-Noether divisor Mrg,d (¯ α) as being the locus of pointed curves [C, q] ∈ Mg,1 possessing a linear series l ∈ Grd (C) with ramiﬁcation sequence αl (q) ≥ α ¯ . It follows from [EH3] that the cone spanned by the pointed BrillNoether divisors on Mg,1 is 2-dimensional, with generators [W g ] and the pull-back of the Brill-Noether class from Mg . Our aim is to analyze the divisor Θg,1 , whose r α), and which seems deﬁnition is arguably simpler than that of the divisors Mg,d (¯ to have been overlooked until now. A consequence of the calculation is that (as expected) [Θg,1 ], lies outside the Brill-Noether cone of Mg,1 . We begin by recalling basic facts about divisors on Mg,1 . For i = 1, . . . , g − 1, the divisor Δi on Mg,1 is the closure of the locus of pointed curves [C ∪D, q], where C and D are smooth curves of genus i and g − i respectively, and q ∈ C. Similarly, Δirr denotes the closure in Mg,1 of the locus of irreducible 1-pointed stable curves. We set δi := [Δi ], δirr := [Δirr ] ∈ Pic(Mg,1 ), and recall that ψ ∈ Pic(Mg,1 ) is the universal cotangent class. Clearly, p∗ (δirr ) = δirr and p∗ (δi ) = δi +δg−i ∈ Pic(Mg,1 ) for 1 ≤ i ≤ [g/2]. For g ≥ 3, the group Pic(Mg,1 ) is freely generated by the classes λ, ψ, δirr , δ1 , . . . , δg−1 , cf. [AC1]. When g = 2, the same classes generate Pic(M2,1 ) subject to the Mumford relation λ=

1 1 δirr + δ1 , 10 5

expressing that λ is a boundary class. We expand the class [Θg,1 ] in this basis of Pic(Mg,1 ), g−1 M Θg,1 ≡ aλ + bψ − birr δirr − bi δi ∈ Pic(Mg,1 ), i=1

and determine the coeﬃcients in a classical way, by understanding the restriction of Θg,1 to suﬃciently many geometric subvarieties of Mg,1 . To ease calculations, we set Ng− := 2g−1 (2g − 1) and Ng+ := 2g−1 (2g + 1), to be the number of odd (respectively even) theta-characteristic on a curve of genus g.

Brill-Noether geometry on moduli spaces of spin curves

273

We deﬁne some test-curves in the boundary of Mg,1 . For an integer 2 ≤ i ≤ g − 1, we choose general (pointed) curves [C] ∈ Mi and [D, x, q] ∈ Mg−i,2 . In particular, we may assume that x, q ∈ D do not appear in the support of any − + odd theta-characteristic ηD on D, and that h0 (D, ηD ) = 0, for any even theta+ characteristic ηD . By joining C and D at a variable point x ∈ C, we obtain a family of 1-pointed stable curves Fg−i := {[C ∪x D, q] : x ∈ C} ⊂ Δg−i ⊂ Mg,1 , where the marked point q ∈ D is ﬁxed. It is clear that Fg−i · δg−i = 2 − 2i, Fg−i · λ = Fg−i · ψ = 0. Moreover, Fg−i is disjoint from all the other boundary divisors of Mg,1 . + Proposition 4.1. For each 2 ≤ i ≤ g − 1, one has that bg−i = Ni− · Ng−i /2. −

− + Proof. We observe that the curve Fg−i ×Mg,1 S g splits into Ni+ · Ng−i + Ni− · Ng−i irreducible components, each isomorphic to C, corresponding to a choice of a pair of theta-characteristics of opposite parities on C and D respectively. Let t ∈ with underlying stable curve Fg−i · Θg,1 be an arbitrary 7 point in the intersection, G C ∪x D, and spin curves [C, ηC ], [D, ηD ] ∈ Si × Sg−i on the two components. + − Suppose ﬁrst that ηC = ηC and ηD = ηD , that is, t corresponds to an even theta-characteristic on C and an odd theta-characteristic on D.7 Then there existG 7 G + − non-zero sections σC ∈ H 0 C, ηC ⊗ OC ((g − i)x) and σD ∈ H 0 D, ηD ⊗ OD (ix) such that (7) ordx (σC ) + ordx (σD ) ≥ g − 1, and σD (q) = 0.

In other words, σC and σD are the aspects of a limit g0g−1 on C ∪x D which vanishes at q ∈ D. Clearly, ordx (σC ) ≤ g − i − 1, hence div(σD ) ≥ ix + q, that − is, q ∈ supp(ηD ). This contradicts the generality assumption on q ∈ D, so this situation does not occur. − + Thus, we are left to consider the case ηC = ηC and ηD = ηD . We denote again + − 0 0 by σC ∈ H (C, ηC ⊗ OC ((g − i)x)) and σD ∈ H (D, ηD ⊗ OD (ix)) the sections + satisfying the compatibility relations (7). The condition h0 (D, ηD ⊗OD (x−q)) ≥ 1 deﬁnes a correspondence on D × D, cf. [DK], in particular, we can choose the + points x, q ∈ D general enough such that H 0 (D, ηD ⊗ OD (x − q)) = 0. Then ordx (σD ) ≤ i − 2, thus ordx (σC ) ≥ g − i + 1. It follows that we must have equality − ordx (σC ) = g − i + 1, and then, x ∈ supp(ηC ). An argument along the lines of [EH3] Lemma 3.4, shows that each of these intersection points has to be counted with multiplicity 1, thus − + ) · Ni− · Ng−i . Fg−i · Θg,1 = #supp(ηC

We conclude by noting that (2i − 2)bg−i = Fg−i · Θg,1 . Proposition 4.2. The relation b = Ng− /2 holds. Proof. Having ﬁxed a general curve [C] ∈ Mg , by considering the ﬁbre p∗ ([C]) inside the universal curve, one writes the identity (2g − 2)b = p∗ ([C]) · Θg,1 = (g − 1)Ng− .

274

G. Farkas

We compute the class of the restriction of the divisor Θg,1 over Mg,1 : Proposition 4.3. One has the linear equivalence Θg,1 ≡ Ng−

71 1 G ψ + λ ∈ Pic(Mg,1 ). 2 4

Proof. We consider the universal pointed spin curve − pr : S− g,1 := Sg ×Mg Mg,1 → Mg,1 .

As usual, P ∈ Pic(S− g,1 ) denotes the universal spin bundle, which over the stack , is a root of the dualizing sheaf ωpr , that is, 2c1 (P) = pr∗ (ψ). We introduce S− g,1 the divisor − − Z := {[C, η, q] ∈ Sg,1 : q ∈ supp(η)} ⊂ Sg,1 , 7 ∗ G and clearly Θg,1 := pr∗ (Z). We write [Z] = c1 (P) − c1 pr (pr∗ (P)) , and take into account that c1 (pr! (P)) = 2c1 (pr∗ (P)) = −λ/2. The rest follows by applying the projection formula. In order to determine the remaining coeﬃcients b0 , b1 , we study the pull-back of Θg,1 under the map ν : M1,2 → Mg,1 , given by ν([E, x, q]) := [C ∪x E, q] ∈ Mg,1 , where [C, x] ∈ Mg−1,1 is a ﬁxed general pointed curve. On the surface M1,2 , if we denote a general element by [E, x, q], one has the following relations between divisor classes, see [AC2]: ψx = ψq , λ = ψx − δ0:xq , δirr = 12(ψx − δ0:xq ). We describe the pull-back map ν ∗ : Pic(Mg,1 ) → Pic(M1,2 ) at the level of divisors: ν ∗ (λ) = λ, ν ∗ (ψ) = ψq , ν ∗ (δirr ) = δirr , ν ∗ (δ1 ) = −ψx , ν ∗ (δg−1 ) = δ0:xq . By direct calculation, we write ν ∗ (Θg,1 ) ≡ (a + b − 12b0 + b1 )ψx − (a + bg−1 − 12b0 )δ0:xq . We compute b0 and b1 by describing ν ∗ (Θg,1 ) viewed as an explicit divisor on M1,2 : − Proposition 4.4. One has the relation ν ∗ (Θg,1 ) ≡ Ng−1 · T2 ∈ Pic(M2,1 ), where

T2 := {[E, x, q] ∈ M1,2 : 2x ≡ 2q}. Proof. We ﬁx an arbitrary point t := [C ∪x E, q] ∈ ν ∗ (Θg,1 ). Suppose ﬁrst that E is a smooth elliptic curve, that is, j(E) = K ∞ and x = K q. Then there exist theta-characteristics of opposite parities ηC , ηE on C and E respectively, together with non-zero sections 7 G 7 G σC ∈ H 0 C, ηC ⊗ OC (x) and σE ∈ H 0 E, ηE ⊗ OE ((g − 1)x) , such that σE (q) = 0 and ordx (σC ) + ordx (σE ) ≥ g − 1. + − + First we assume that ηC = ηC and ηE = ηE , thus, ηE = OE . Since H 0 (C, ηC )= 0, one obtains that ordx (σC ) = 0, that is ordx (σE ) = g − 1, which is impossible,

Brill-Noether geometry on moduli spaces of spin curves

275

because σE must vanish at q as well. Thus, one is led to study the remaining case, − + − when ηC = ηC and ηE = ηE . Since x ∈ / supp(ηC ), we obtain ordx (σC ) ≤ 1, and then by compatibility, the last inequality becomes equality, while ordx (σE ) = g −2, + − hence ηE = OE (x − q), or equivalently, [E, x, q] ∈ T2 . The multiplicity Ng−1 in − , the expression of ν ∗ (Θg,1 ) comes from the choices for the theta-characteristics ηC responsible for the C-aspect of a limit g0g−1 on C ∪x E. It is an easy moduli count to show that the cases when j(E) = ∞, or [E, x, q] ∈ δ0:xq (corresponding to the situation when x and q coalesce on E), do not occur generically on a component of ν ∗ (Θg,1 ). Proposition 4.5. T2 is an irreducible divisor on M1,2 of class T2 ≡ 3ψx ∈ Pic(M1,2 ). Proof. We write T2 ≡ αψx − βδ0:xq ∈ Pic(M1,2 ), and we need to understand the intersection of T2 with two test curves in M1,2 . First, we ﬁx a general point [E, q] ∈ M1,1 and consider the family E1 := {[E, x, q] : x ∈ E} ⊂ M1,2 . Clearly, E1 · δ0:xq = E1 · ψx = 1. On the other hand E1 · T2 is a 0-cycle simply supported at the points x ∈ E − {q} such that x − q ∈ Pic0 (E)[2], that is, E1 · T2 = 3. This yields the relation α − β = 3. As a second test curve, we denote by [L, u, x, q] ∈ M0,3 the rational 3-pointed rational curve, and deﬁne the pencil R := {[L ∪u Eλ , x, q] : λ ∈ P1 } ⊂ M1,2 , where {Eλ }λ∈P1 is a pencil of plane cubic curves. Then R ∩T2 = ∅. Since R ·λ = 1 and R · δirr = 12, we obtain the additional relation β = 0, which completes the proof. Putting together Propositions 4.1, 4.3 and 4.5, we obtain the system of equations − a + bg−1 − 12birr = 0, a − 12birr + b + b1 = 3Ng−1 , a=

1 − 1 3 − Ng , b = Ng− , b1 = Ng−1 . 4 2 2

Thus birr = 22g−6 and bg−1 = 2g−3 (2g−1 +1). This completes the proof of Theorem 0.3.

References [AC1] E. Arbarello and M. Cornalba, The Picard groups of the moduli space of curves, Topology 26 (1987), 153-171. [AC2] E. Arbarello and M. Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Etudes Sci. Publ. Math. 88 (1998), 97-127.

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I. Bauer and A. Verra, The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian, arXiv:0808.1318, Michigan Math. Journal (2010), to appear.

[C]

M. Cornalba, Moduli of curves and theta-characteristics, in: Lectures on Riemann surfaces (Trieste, 1987), World Scientiﬁc Press 1989, 560-589.

[Cu]

F. Cukierman, Families of Weierstrass points, Duke Mathematical Journal 58 (1989), 317-346.

[DK]

I. Dolgachev and V. Kanev, Polar covariants of plane cubics and quartics, Advances in Mathematics 98 (1993), 216-301.

[EH1] D. Eisenbud and J. Harris, Limit linear series: Basic theory, Inventiones Math. 85 (1986), 337-371. [EH2] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus 23 Inventiones Math. 90 (1987), 359–387. [EH3] D. Eisenbud and J. Harris, Irreducibility of some families of linear series with ´ Brill-Noether number −1, Annales Scientiﬁque Ecole Normale Sup´erieure 22 (1989), 33-53. [F1]

G. Farkas, Koszul divisors on moduli spaces of curves, American Journal of Mathematics 131 (2009), 819-869.

[F2]

G. Farkas, The birational type of the moduli space of even spin curves, Advances in Mathematics 223 (2010), 433-443.

[FV]

G. Farkas and A. Verra, The intermediate type of certain moduli spaces of curves, arXiv:0910.3905.

[FL]

G. Farkas and K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties, Journal of the European Mathematical Society 12 (2010), 755-795.

[FMP] G. Farkas, M. Mustat¸˘ a and M. Popa, Divisors on Mg,g+1 and the Minimal Resolu´ tion Conjecture for points on canonical curves, Annales Scientiﬁque Ecole Normale Sup´erieure 36 (2003), 553-581. [HM]

J. Harris and D. Mumford, On the Kodaira dimension of Mg , Inventiones Math. 67 (1982), 23-88.

[Kh]

D. Khosla, Tautological classes on moduli spaces of curves with linear series and a push-forward formula when ρ = 0, arXiv:0704.1340.

[L]

R. Lazarsfeld, A sampling of vector bundle techniques in the study of linear systems, in: Lectures on Riemann Surfaces (Trieste 1987), World Scientiﬁc Press 1989, 500-559.

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A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, American Journal of Math. 125 (2003), 105-138.

[Lud] K. Ludwig, On the geometry of the moduli space of spin curves, Journal of Algebraic Geometry 19 (2010), 133-171. [R]

M. Raynaud, Sections des ﬁbr´es vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103-125.

Gavril Farkas: Humboldt-Universit¨ at zu Berlin, Institut f¨ ur Mathematik, Unter den Linden 6, D 10099 Berlin, Germany E-mail: [email protected]

On the Bimeromorphic Geometry of Compact Complex Contact Threefolds Kristina Frantzen and Thomas Peternell

Abstract. We prove that a compact contact threefold which is bimeromorphically equivalent to a K¨ ahler manifold and not rationally connected is the projectivised tangent bundle of a K¨ ahler surface.

Contents 1 Introduction

277

2 Uniruledness and splitting

278

3 The case of a 2-dimensional rational quotient

280

4 The case of a 1-dimensional rational quotient

285

1. Introduction A (compact) complex manifold X of dimension 2n+1 is a contact manifold if there exists a vector bundle sequence θ

0 → F → TX → L → 0,

(v)

where TX is the tangent 42bundle and L a line bundle, with the additional property that the induced map F → L, v ∧ w ?→ [v, w]/F ∈ L, given by the Lie bracket [ , ] on TX is everywhere non-degenerate. The line bundle L is referred to as the contact line bundle on X. There are two basic ways to construct contact structures. •

A simple Lie group gives rise to a Fano contact manifold X (with b2 (X) = 1) by taking the unique closed orbit for the adjoint action of the Lie group on the projectivised Lie algebra, see e.g. [Bea98].

•

For any compact complex manifold Y the projectivised tangent bundle X = P(TY ) is a contact manifold.

Now the question naturally arises whether any compact complex contact manifold is given in this way.

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In the following, let X be a compact complex contact manifold. If X is projective with b2 (X) = 1, then X must be a Fano manifold and Beauville [Bea98] proved partial results towards the realisation as closed orbit. In general, if X is K¨ ahler, Demailly [Dem02] showed that the canonical bundle KX is not nef. If X is projective with b2 (X) ≥ 2, Theorem 1.1 in [KPSW00] provides a positive answer to the question above. If X is K¨ ahler but not projective, then necessarily b2 (X) ≥ 2 and the second alternative is conjectured to hold, i.e., X should be a projectivised tangent bundle. However the paper [KPSW00] essentially uses Mori theory, which is, at the moment, not available in the K¨ ahler case, except in dimension 3 where it can be shown that X is a projectivised tangent bundle over a surface (see Section 2). In this paper we go one step further in dimension 3: we consider contact threefolds X which are in class C, i.e., bimeromorphic to a K¨ahler manifold. We ﬁrst show that these threefolds must be uniruled. Then we consider the rational quotient r : X ''( Q. The meromorphic map r identiﬁes two very general points if and only if they can be joined by a chain of rational curves. In particular, X is rationally connected if and only if dim Q = 0. We distinguish the cases dim Q = 1 (Theorem 4.5) and dim Q = 2 (Theorem 3.7) and show Theorem. Let X be a compact contact threefold in class C. Assume that X is not rationally connected. Then there exists a K¨ ahler surface Y such that X @ P(TY ). In particular, X is K¨ ahler. The remaining open case that X is rationally connected, in particular Moishezon, will require diﬀerent methods. Probably it will be necessary to consider rational curves C with −KX · C minimal, but positive.

2. Uniruledness and splitting We shall use the following notation: Deﬁnition 2.1. A compact complex manifold X is said to be in class C if X is bimeromorphically equivalent to a K¨ ahler manifold. An important property of manifolds in class C is the compactness of the irreducible components of the cycle space (cf. [Cam80]). The key for our investigations is the following Theorem 2.2. Let X be a compact contact threefold in class C. Then X is uniruled. ˆ → X be a modiﬁcation such that X ˆ is K¨ahler. It is a wellProof. Let π : X ˆ established fact that X is uniruled if and only if KXˆ is not pseudo-eﬀective, i.e., the Chern class c1 (KXˆ ) is not represented by a positive closed current. The projective case in any dimension is treated in [BDPP04] based on the uniruledness theorem of Miyaoka-Mori [MM86]. The K¨ ahler case in dimension three has been proved by Brunella [Bru06, Cor. 1.2].

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279

The contact structure on X is given by θ ∈ H 0 (X, Ω1X ⊗L); note that θ∧dθ K= 0. ˆ Ω1 ⊗ π ∗ (L)). By [Dem02, Via π ∗ , the L-valued form θ induces a form θˆ ∈ H 0 (X, ˆ X ∗ −1 Cor. 1], the pullback of the dual line bundle π (L ) is not pseudo-eﬀective. Since KX = 2L−1 , the line bundle π ∗ (KX ) is not pseudo-eﬀective which is equivalent to saying that KX is not pseudo-eﬀective. Since π∗ (KXˆ ) = KX , the Chern class c1 (KXˆ ) cannot be represented by a positive closed current Tˆ, because otherwise c1 (KX ) would be represented by the positive closed current π∗ (Tˆ). Hence KXˆ is not pseudo-eﬀective, and we conclude by the uniruledness criterion stated above. As a consequence we obtain the following classiﬁcation result for compact K¨ahler contact threefolds generalising the well-known projective case (we refer to [KPSW00] for further references). Corollary 2.3. Let X be a compact K¨ ahler contact threefold. Then either X @ P3 or X = P(TY ) for a K¨ ahler surface Y . Proof. By Theorem 2.2 above, the threefold X is uniruled. In particular, there is a positive-dimensional subvariety through the general point of X, i.e., X is not simple. The claim now follows from [Pet01, Theorem 4.1]. Remark 2.4. The proof of Theorem 2.2 above actually shows that the canonical bundle of a compact contact manifold in class C of any dimension is not pseudoeﬀective. However in dimensions at least 4, unless X is projective, it is completely open, whether this implies uniruledness. We now make a digression and consider the contact sequence (v). It is an interesting question whether this sequence can split. In the case where X is Fano, LeBrun [LeB95, Cor. 2.2] showed that splitting never occurs. By the following theorem, the same is true if X is in class C. Theorem 2.5. Let X be a compact contact manifold in class C. Then the contact sequence (v) does not split. Proof. Suppose we have a splitting TX @ F ⊕ L, hence Ω1X @ F ∗ ⊕ L∗ . Then it is well-known (see e.g. Beauville [Bea00]), that c1 (L) ∈ H 1 (X, L∗ ) ⊂ H 1 (X, Ω1X ) and c1 (F ) ∈ H 1 (X, F ∗ ). Since c1 (F ) = nc1 (L) in H 1 (X, Ω1X ) and since X is in class C, we conclude that c1 (L) = c1 (F ) = 0 in H 1 (X, Ω1X ), and therefore also in H 2 (X, R). Hence KX is numerically trivial and consequently, due to Remark 2.4, X cannot be in class C. Let X be a compact contact manifold X with contact sequence (v). A subvariety S ⊂ X, i.e., a closed irreducible analytic subset in X, is called F -integral if TS,x ⊂ Fx for all smooth points x ∈ S. Notation. A holomorphic family (Ct )t∈T of curves in X is given by a diagram Z # T

q

p

!X

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Kristina Frantzen and Thomas Peternell

such that •

T is an irreducible subspace of the cycle space of curves in X

•

q −1 (t) is the cycle corresponding to t ∈ T

•

Ct = p(q −1 (t)) as cycle.

An important tool will be the following lemma, the proof of which is based on the observation that a surface covered by a family of F -integral curves is itself F -integral. Lemma 2.6. Let X be a compact contact threefold with contact line bundle L. Let (Ct )t∈T be a 1-dimensional family of generically irreducible rational curves passing through a ﬁxed point x0 ∈ X. Then L · Ct ≥ 2. Proof. We assume to the contrary that L · Ct ≤ 1 for all t ∈ T . By restricting the contact sequence (v) to an irreducible rational curve Ct and, if necessary, pulling it back to the normalisation η : C˜t → Ct , one ﬁnds that the map O(2) @ TC˜t → η ∗ L @ O(a) with a ≤ 1 is trivial and therefore TCt ,x n→ Fx for x ∈ Ct smooth. I.e., for the general t ∈ T the curve Ct is F -integral. : Consider the surface S = t∈T Ct ⊂ X covered by the curves Ct . Then the proof of [Keb01, Proposition 4.1] shows that S is F -integral. But since any F integral subvariety in X has dimension at most 1, this yields a contradiction.

3. The case of a 2-dimensional rational quotient We assume in this section that X is a compact contact threefold in class C with a rational quotient r : X ''( Q of dimension dim Q = 2. We refer the reader to the books [Deb01], [Kol96] and the references therein for relevant details on the contruction and the properties of a rational quotient. A fundamental result of Graber, Harris, and Starr [GHS03] states that the quotient Q is not uniruled. In case X has dimension three, this result actually has previously been known. The meromorphic map r : X ''( Q is almost holomorphic, i.e., r is proper holomorphic on a dense open set in X, and its general ﬁber is P1 . Thus, we have a unique covering family (lt )t∈T of rational curves with graph Z, Z # T

q

p

!X ' 'r ' # Q

and dim T = 2. Since r is almost holomorphic, the map p is bimeromorphic. By possibly passing to the normalisation, we may assume both Z and T normal. Moreover, we may take Q = T .

Bimeromorphic Geometry of Contact Threefolds

281

Lemma 3.1. All curves lt satisfy L · lt = 1 and all irreducible curves lt are F integral. Proof. The general lt is a general ﬁber of r, hence −KX · lt = 2 by adjunction and therefore L · lt = 1. The same is then true for all lt . All irreducible curves lt are consequently F -integral (cf. proof of Lemma 2.6). In the following we will make use of the deformation theory of rational curves. This is to say we consider a rational curve C ⊂ X, given by a bimeromorphic morphism f : P1 → X and consider the deformations ft of f . We obtain a family (Ct ) = (ft (P1 )) and then take its closure in the cycle space, because in general the family (ft ) will not be compact, or in other words, the family (Ct ) will split. Here it is essential that X is in class C, hence all irreducible components of the cycle space of X are compact. We repeatedly use the following basic fact, see e.g. [Kol96, Theorem II.1.3]. Fact 3.2. Let X be a compact threefold and let C be a rational curve in X. If −KX · C ≥ m, then C will deform as rational curve in an at least m-dimensional family. The following proposition is the technical core of this section. Proposition 3.3. The map p : Z → X is an isomorphism. In particular, the rational quotient r : X → T is holomorphic and equidimensional. Proof. For x ∈ X we let T (x) be the analytic subset of all t ∈ T such that x ∈ lt . Since the general lt does not pass through x, it follows that dim T (x) ≤ 1. In the following we show that dim T (x) = 0 for all x; in other words, p is ﬁnite. Since the map p is of degree 1 and has connected ﬁbers by Zariski’s main theorem, the ﬁniteness of p forces p to be biholomorphic. Suppose now to the contrary that dim T (x) = 1 for some ﬁxed x ∈ X. If T (x) happens to be reducible, we replace it by an irreducible component of dimension 1. In the following, we shall therefore assume that T (x) is irreducible. Let S be the surface covered by the lt belonging to T (x): ' S= lt . t∈T (x)

If the general lt through x is irreducible, then the Lemmata 2.6 and 3.1 yield a contradiction. So we are left with the case when all lt , t ∈ T (x) are reducible. In this case S itself might be reducible.%For t ∈ T (x) we decompose lt into its irreducible components and write lt = ajt Ctj . Since L · lt = 1 for all t ∈ T , there exists at j least one component Ct in this decomposition with L · Ctj ≥ 1. We pick t ∈ T (x) general and let C (1) be a component of lt with L · C (1) ≥ 1. Then by Fact 3.2, (1) C (1) deforms in an at least 2-dimensional family (Ct )t∈T1 . Suppose ﬁrst that (1) L · Ct = 1.

282

Kristina Frantzen and Thomas Peternell (1)

If the family (Ct )t∈T1 covers a surface, then we ﬁnd a 1-dimensional subfamily through a ﬁxed point, contradicting Lemma 2.6. If the family covers all of X, then, since there is only one covering family of generically irreducible rational curves in (1) X, the family (Ct )t∈T1 must be the original family (lt )t∈T , in particular T = T1 . In other words, we have t0 ∈ T (x) and t1 ∈ T such that lt0 = lt1 + R with an eﬀective curve R. Thus p−1 (x) contains more than one point for every x ∈ lt1 . Since p has connected ﬁbers, we conclude that dim p−1 (x) = 1 for every x ∈ lt1 . Then either all curves lt , t ∈ T pass through lt1 , which is absurd since r is almost holomorphic, or there exists a 1-dimensional subfamily (lt1 +Cu )u∈U of (lt )t∈T with dim U = 1. In this second case however, since the subfamily (lt1 + Cu )u∈U does not contain the curve lt1 itself, it follows p−1 (x) is not connected, a contradiction. So we are left with (1) L · Ct ≥ 2, (1)

(1)

i.e., −KX · Ct ≥ 4, and the Ct deform in an at least 4-dimensional family (cf. Fact 3.2) which must stay in an irreducible component S1 of S. (The deformations (1) of Ct cannot cover all of X since there is a unique covering family of generically irreducible rational curves in X, and this family, namely (lt )t∈T , is 2-dimensional (2) and fulﬁls −KX · lt = 2). We want to exhibit a new family (Ct ) in the surface (2) S1 such that L · Ct ≥ 2. In order to construct this new family we notice that the 4-dimensional family (1) (Ct )t∈T1 must split. In fact, through any two points of S1 there is a positivedimensional subfamily. Now we choose carefully a splitting component C (2) such that L · C (2) ≥ 2, namely we want to achieve that C (2) passes through a general point of S1 . By Lemma 3.5 we obtain a generically non-splitting family (hu )u∈U of rational curves hu in S1 with dim U ≥ 2 such that for general u ∈ U there exists (1) t(u) ∈ T1 such that hu is an irreducible component of Ct(u) . Since the family (hu ) covers exactly S1 , there exists a 1-dimensional subfamily through a general point of S1 and we take C (2) to be a general member of this subfamily. By Lemma 2.6 we obtain L · C (2) ≥ 2. Again, −KX · C (2) ≥ 4 implies that C (2) moves in an at least 4-dimensional family (2) of rational curves, say (Ct )t∈T2 . (k) Inductively we obtain families (Ct )t∈Tk in S1 such that (k)

L · Ct

≥ 2.

Our aim now is to ﬁnd an argument that this procedure must stop at some point, (k) i.e., that L · Ct ≥ 2 cannot occur inﬁnitely many times. If X is K¨ ahler with K¨ ahler form ω, this follows from the fact that the intersec(k) (k) tion number Ct · ω strictly decreases and that all classes Ct are integer classes in a ball {a | a · ω ≤ K} ⊂ H 2 (X, R). Let us brieﬂy explain the diﬃculty arising from the fact that X is not necessarily K¨ ahler. If X is merely in class C, we cannot argue in this way, because we will

283

Bimeromorphic Geometry of Contact Threefolds

have curves with “semi-negative” cohomology. To be precise, we choose a sequence ˆ → X such that X ˆ is K¨ of blow-ups in points and smooth curves π : X ahler, ﬁx ˆ a K¨ ahler form ω ˆ on X, and form the current R = π∗ (ω). Then R · C > 0 for all curves not contained in the center of π. On the other hand, there are ﬁnitely many curves B1 , . . . , BN such that R · Bj ≤ 0. These “bad” curves have to be taken into account. We are able to get around this diﬃculty since every splitting takes place in the ﬁxed surface S1 . Inside this surface we will not have any curves with “negative” homology. We consider the normalisation η : S˜1 → S1 . (k) (k) (k) We deﬁne a family (C˜t ) in S˜1 by letting C˜t be the strict transform of Ct in (k) S˜1 for general t and then taking the closure in the cycle space. Let C˜ be the strict transform of C (k) in S˜1 . Then we obtain a splitting (k) ˜k C˜tk = C˜ (k+1) + R

for some tk . Inductively we ﬁnd (m) ˜m, C˜ (m) ≡ C˜tm = C˜ (m+1) + R

for all m ∈ N. Here ≡ denotes homology equivalence in S˜1 . It follows that C˜ (1) ≡ C˜ (m) +

m−1 M

˜j , R

j=1

i.e., C˜ (1) is homology equivalent to a sum of arbitrarily many eﬀective curves in S˜1 . This contradicts Lemma 3.4. We now prove the two technical lemmata used in the proof of Proposition 3.3 above. Lemma 3.4. Let S be a compact connected normal Moishezon surface. Then there exists a linear map ϕ : H2 (S, Q) → Q such that ϕ([C]) ≥ 1 for all classes of irreducible curves C in S. Proof. It suﬃces to construct ϕ on the subspace V of H2 (S; Q) generated by classes of irreducible curves. Let σ : Sˆ → S be a desingularisation of S and note that the ˆ be an ample divisor on Sˆ and σ∗ (H) ˆ = H be its surface Sˆ is projective. We let H push-down to S. Using the intersection theory on normal surfaces established in [Mum61] and [Sak84], we deﬁne ϕ([C]) = C · H = (σ ∗ C) · (σ ∗ H). % Here σ ∗ D denotes the sum D + ai Ei of the strict transform D of the divisor D in Sˆ and an appropriately weighted sum of the exceptional curves of σ.

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Kristina Frantzen and Thomas Peternell

In order to check that ϕ is well-deﬁned on homology classes, it suﬃces to show that c1 (O(σ ∗ C)) = 0 for every C with [C] = 0 ∈ H2 (S, Q). Following the notation and results presented in [Sak84], Section 3, this is equivalent to c1 (O(C)) ∈ ker(σ ∗ ) = ker(ηS ) ⊂ H 2 (S, Q). Here ηS : H 2 (S, Q) → H2 (S, Q) denotes the Poincar´e homomorphism on S. We may write 0 ηS (c1 (O(C))) = σ∗ (ηSˆ (c1 (O(C)))) 0 = ˆ Q) → H2 (S, ˆ Q) on Sˆ and O(C) for the Poincar´e isomorphism ηSˆ : H 2 (S, ∗ ∗ ∗ O(σ C). Since ηSˆ (c1 (O(σ C))) = [σ C] on the smooth surface S, we conclude ηS (c1 (O(C))) = σ∗ [σ ∗ C] = [C] = 0 and obtain the desired vanishing. It remains to check that ϕ([C]) ≥ 1 for all classes of irreducible curves C in S. We have M ϕ([C]) = (σ ∗ C) · (σ ∗ H) = (σ ∗ C) · (H + bi Ei ). Since (σ ∗ C) · Ej = 0 for all j by deﬁnition of σ ∗ C (cf. [Sak84, Section 1]), we conclude M ai Ei ) · H. ϕ([C]) = (σ ∗ C) · H = (C + ˆ is ample and C is eﬀective, in particular C is eﬀective and ai > 0 for Since H = H all i, the desired inequality follows. Lemma 3.5. Let S be an irreducible Moishezon surface with a covering family (Ct )t∈T of (rational) curves. Suppose dim T ≥ 4. Let T D ⊂ T be the subset of those t for which Ct splits. Then dim T D ≥ 2. Proof. Let x ∈ S and T (x) = {t ∈ T | x ∈ Ct }. Since dim T ≥ 4 by assumption, we have dim T (x) ≥ 3. (Consider the graph p : Z → S of the family (Ct )t∈T and observe that dim(p−1 (x)) ≥ 3 and q : Z → T restricted to p−1 (x) is ﬁnite.) The same dimension count substituting T by T (x) shows that dim(T (x) ∩ T (xD )) ≥ 2 for x, xD ∈ S. Hence there exists a 2-dimensional subfamily through x, xD and therefore we obtain a 1-dimensional subfamily through x and xD such that all members split. In other words dim(T D ∩ T (x)) ≥ 1. Varying x we conclude dim T D ≥ 2. Having established Proposition 3.3, it remains to show that the rational quotient r : X → T is actually a P1 -bundle. Proposition 3.6. Assume that the rational quotient r : X → T is holomorphic and equidimensional. Then r is a P1 -bundle, T is smooth and X = P(TT ).

Bimeromorphic Geometry of Contact Threefolds

285

Proof. As a ﬁrst step, we show that the ﬁbers of r must be irreducible. Assume the contrary and let r−1 (t0 ) = lt0 = C (1) +R be a reducible ﬁber such that L·C (1) ≥ 1. Then C (1) deforms in an at least 2-dimensional family, hence C (1) is a member of (lt )t∈T , i.e., C (1) = lt1 for a suitable t1 ∈ T . Since r is holomorphic, this is only possible when t0 = t1 , a contradiction. So r : X → T is a holomorphic, equidimensional map of normal complex spaces and every ﬁber of r is a reduced, irreducible rational curve. Now the arguments of [Kol96, Theorem II.2.8] can be adapted to our situation and it follows that r is a P1 -bundle. In particular, T is smooth and X = P(TT ). Recall that a surface in class C is K¨ ahler. In total we have shown: Theorem 3.7. Let X be a compact contact threefold in class C. If the rational quotient has dimension 2, then X is K¨ ahler and of the form P(TY ) with a K¨ ahler surface Y . The projection X → Y is the rational quotient, i.e., Y is not uniruled.

4. The case of a 1-dimensional rational quotient In this section we assume that X is a compact contact threefold in class C with contact line bundle L and a rational quotient r : X ''( Q of dimension dim Q = 1. Then necessarily X is Moishezon and Q is a smooth curve B of genus at least 1. We observe that r : X → B is holomorphic. Our aim is to show that X is of the form X = P(TY ) for some surface Y . The surface Y is then necessarily Moishezon, and since a smooth Moishezon surface is projective, we are going to show directly that X is projective. Let B0 be the set of points b in B such that r−1 (b) = Xb is smooth. Lemma 4.1. Let b ∈ B0 . Then Xb is a Hirzebruch surface Xb = P(OP1 ⊕OP1 (−e)) with e > 0 even. Proof. By adjunction KXb = −2L|Xb , hence Xb is minimal. Moreover, Xb cannot be a projective plane P2 or a Hirzebruch surface with odd e. To exclude the quadric (e = 0), observe that Xb is not F -integral, i.e., the restriction of the contact form θ to Xb does not vanish identically, and hence K 0, θ|Xb ∈ H 0 (Xb , Ω1Xb ⊗ L|Xb ) = which is impossible for Xb = P1 × P1 . Every smooth ﬁber Xb of r has a uniquely deﬁned non-splitting 1-dimensional family of rational curves, namely the ruling lines. All these rational curves together give rise to a 2-dimensional family (ly )y∈Y of rational curves in X, where Y is the irreducible component of the cycle space parametrising generically the ruling lines. We obtain an almost holomorphic map π : X ''( Y and a holomorphic map

286 g : Y → B, g(y) = r(ly ),

Kristina Frantzen and Thomas Peternell

π X ! ! !! Y ⊃ Y0 g

# B ⊃ B0 such that Y0 = g −1 (B0 ) is a P1 -bundle over B0 and π is again a P1 -bundle over Y0 . In fact, if Yb is the ﬁber over b ∈ B0 , then Xb = P(O ⊕ O(−eb )|Yb ). The technical key to the main result of this section is Proposition 4.2. The family (ly )y∈Y does not split. Proof. Suppose (ly ) splits. Then there exists a point y0 ∈ Y , an irreducible curve C (1) with L · C (1) ≥ 1 and an eﬀective curve R such that ly0 = C (1) + R. As seen (1) before the curve C (1) deforms in an at least 2-dimensional family (Ct )t∈T1 . (1) (1) (1) Let us ﬁrst consider the case L · Ct = 1. If the family (Ct )t∈T1 covers a surface, we ﬁnd a 1-dimensional subfamily through a ﬁxed point and contradict (1) Lemma 2.6. If (Ct )t∈T1 covers all of X, we recover the original family (ly ) as (1) follows: notice that the general Ct will be an irreducible rational curve in a smooth ﬁber Xb and −KXb · Ct = 2 by adjunction. This implies that the general (1) curve Ct must be a ruling line, i.e., the general curve Ct must be a curve lt . This may now be excluded using Lemma 2.6 by the same arguments as in Proposition 3.3. (1) (1) (2) Having ruled out L · Ct = 1, we consider the case L · Ct ≥ 2, i.e., (1)

−KX · Ct

(1)

(1)

= 2L · Ct

≥ 4.

(2a) Assume that the family (Ct )t∈T1 covers all of X and choose a general point x ∈ X. A dimension count shows that there is a 2-dimensional subfamily (1) (Ct )t∈T1 (x) through the point x, necessarily ﬁlling a rational surface S, which must be a ﬁber of r. Since x is general, there is a b ∈ B with Xb smooth such that (1) (1) S = Xb . The family (Ct )t∈T1 (x) splits as Ct1 = C (2) + R1 with L · C (2) ≥ 1. If L · C (2) ≥ 2 we repeat the whole process. Assume that L · C (k) ≥ 2 for all k. Then, (1) we obtain a decomposition of the homology class of Ct1 as a sum of arbitrarily %K (1) (k) many eﬀective curves Ct1 ≡ + RK−1 . As we can always choose a k=1 C %K subfamily through a point of S, we can assume that k=1 C (k) + RK−1 ⊂ S for all K. Calculating the degree with respect to an ample line bundle H on S, we obtain (k) a contradiction. Hence at some stage the procedure has to stop, i.e. L · Ct = 1 and we conclude by Lemma 2.6. (1) (2b) It remains to consider the case the case where L · Ct ≥ 2 and the family (1) (Ct ) covers a surface S, which is a component of a ﬁber Xb of r. The family must split and we choose a splitting component C (2) such that L · C (2) ≥ 1. If L · C (2) = 1, we are done again; if L · C (2) ≥ 2, we obtain an at least 4-dimensional (2) family (Ct ). If this family covers X, we are done by the arguments of (2a) applied

287

Bimeromorphic Geometry of Contact Threefolds (2)

(1)

(2)

to (Ct ), instead of (Ct ). Otherwise, the family (Ct ) ﬁlls a component S D of the same ﬁber Xb , and as in the proof of Proposition 3.3, by choosing C (2) carefully, we may assume that S D = S. Now we are in completely the same situation as in the proof of Proposition 3.3 and proceed as described there. In order to apply Proposition 4.2, we consider the normalised graph p : Z → X of the family (ly )y∈Y . p !X Z# ## q ( ## ( ## (π $ %( Y Proposition 4.3. The map p : Z → X is biholomorphic. Proof. The map p is generically biholomorphic, hence by Zariski’s main theorem, it suﬃces to show that p does not have positive-dimensional ﬁbers. So suppose that dim p−1 (x) = 1. Then there exists a 1-dimensional subfamily (ly )y∈Y (x) through x with all ly irreducible by the previous proposition. Since L · ly = 1, we contradict Lemma 2.6. As before in Proposition 3.6 we conclude: Corollary 4.4. The map π : X → Y is a P1 -bundle. We may now apply Lemma 4.6 below to the map π : X → Y and have shown: Theorem 4.5. Let X be a compact contact threefold in class C with 1-dimensional rational quotient B. Then X is projective and there is a smooth projective surface Y with a P1 -ﬁbration Y → B such that X @ P(TY ). Lemma 4.6. Let Y be a complex manifold of dimension n + 1 and π : X → Y be a Pn -bundle. If X is a contact manifold, then X @ P(TY ). Proof. Let Z @ Pn be a ﬁber of π. Adjunction implies that the contact line bundle L restricted to Z fulﬁls L|Z = OZ (1). Setting E = π∗ (L), we conclude (X, L) @ (P(E), OP(E) (1)). Now the last part of the proof of Theorem 2.12 in [KPSW00] can be applied and shows that E @ TY .

References [BDPP04] S´ebastian Boucksom, Jean-Pierre Demailly, Mihai P˘ aun, and Thomas Peternell, The pseudo-eﬀective cone of a compact K¨ ahler manifold of negative Kodaira dimension, arXiv:math/0405285, 2004. [Bea98]

Arnaud Beauville, Fano contact manifolds and nilpotent orbits, Comment. Math. Helv. 73 (1998), no. 4, 566–583.

288

Kristina Frantzen and Thomas Peternell

[Bea00]

, Complex manifolds with split tangent bundle, Complex analysis and algebraic geometry, volume in memory of Michael Schneider, eds. T. Peternell and F.O. Schreyer, de Gruyter, Berlin, 2000, pp. 61–70.

[Bru06]

Marco Brunella, A positivity property for foliations on compact K¨ ahler manifolds, Internat. J. Math. 17 (2006), no. 1, 35–43.

[Cam80]

Frederic Campana, Alg´ebricit´e et compacit´e dans l’espace des cycles d’un espace analytique complexe, Math. Ann. 251 (1980), no. 1, 7–18.

[Deb01]

Olivier Debarre, Higher-dimensional algebraic geometry, Springer-Verlag, New York, 2001.

[Dem02]

Jean-Pierre Demailly, On the Frobenius integrability of certain holomorphic pforms, Complex geometry (G¨ ottingen, 2000), volume in honour of H. Grauert, eds. I. Bauer et al., Springer, Berlin, 2002, pp. 93–98.

[GHS03]

Tom Graber, Joe Harris, and Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67.

[Keb01]

Stefan Kebekus, Lines on contact manifolds, J. Reine Angew. Math. 539 (2001), 167–177.

[Kol96]

J´ anos Koll´ ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996.

Universitext,

[KPSW00] Stefan Kebekus, Thomas Peternell, Andrew J. Sommese, and Jarosxlaw A. Wi´sniewski, Projective contact manifolds, Invent. Math. 142 (2000), no. 1, 1–15. [LeB95]

Claude LeBrun, Fano manifolds, contact structures, and quaternionic geometry, Internat. J. Math. 6 (1995), no. 3, 419–437.

[MM86]

Yoichi Miyaoka and Shigefumi Mori, A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), no. 1, 65–69.

[Mum61]

David Mumford, The topology of normal singularities of an algebraic surface ´ and a criterion for simplicity, Inst. Hautes Etudes Sci. Publ. Math. (1961), no. 9, 5–22.

[Pet01]

Thomas Peternell, Towards a Mori theory on compact K¨ ahler threefolds. III, Bull. Soc. Math. France 129 (2001), no. 3, 339–356.

[Sak84]

Fumio Sakai, Weil divisors on normal surfaces, Duke Math. J. 51 (1984), no. 4, 877–887.

Kristina Frantzen, Mathematisches Institut, Universit¨ at Bayreuth, 95440 Bayreuth, Germany E-mail: [email protected] Thomas Peternell, Mathematisches Institut, Universit¨ at Bayreuth, 95440 Bayreuth, Germany E-mail: [email protected]

Introduction to the theory of quasi-log varieties Osamu Fujino∗

Abstract. This paper is a gentle introduction to the theory of quasi-log varieties by Ambro. We explain the fundamental theorems for the log minimal model program for log canonical pairs. More precisely, we give a proof of the base point free theorem for log canonical pairs in the framework of the theory of quasi-log varieties. 2010 Mathematics Subject Classiﬁcation. Primary 14E30; Secondary 14C20, 14F17. Keywords. Base point free theorem, Minimal model program, Vanishing theorem.

1. Introduction The aim of this article is to explain the fundamental theorems for the log minimal model program for log canonical pairs. More explicitly, we describe the base point free theorem for log canonical pairs in the framework of the theory of quasi-log varieties (see Corollary 4.2). We also treat the cone theorem for log canonical pairs (see Theorem 5.3). This paper is a gentle introduction to Ambro’s theory of quasi-log varieties (cf. [A]). It contains no new statements. However, it must be valuable because there are no introductory articles for the theory of quasilog varieties. The original article [A] seems to be inaccessible even for experts. We basically follow Ambro’s arguments (see [A, Section 5]) but we change them slightly to clarify the basic ideas and to remove some ambiguities and mistakes. The book [F7] contains a comprehensive survey of the fundamental theorems of the log minimal model program from the viewpoint of the theory of quasi-log varieties. A new approach to the log minimal model program for log canonical pairs without using quasi-log varieties was found in [F8]. It seems to be more natural and much easier than the theory of quasi-log varieties. The paper [F9] contains all the details of this new approach and is almost self-contained. Note that we only use Q-divisors for simplicity. Some of the results can be generalized for R-divisors with a little care. We do not treat the relative versions of the fundamental theorems in order to make our arguments transparent. There are no diﬃculties for the reader to obtain the relative versions once he understands this paper. We hope that this article will make the theory of quasi-log varieties more accessible. Note that the reader does not have to refer to [A] in order to read this article. Our formulation is slightly diﬀerent from the one in [A]. So, if the reader wants to taste the original ﬂavor of the theory of quasi-log varieties, then he has to see [A]. ∗ The

author was partially supported by the Grant-in-Aid for Young Scientists (A) 320684001 from JSPS and by the Inamori Foundation.

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We summarize the contents of this paper. In Section 2, we quickly review the torsion-freeness and the vanishing theorem in [F7, Chapter 2]. In Section 3, we introduce the notion of qlc pairs, which is a special case of Ambro’s quasi-log varieties, and prove some important and useful lemmas. Theorem 3.6 is a key result in the theory of quasi-log varieties. Section 4 is devoted to the proof of the base point free theorem for qlc pairs. This section is the heart of this paper. In Section 5, we treat the rationality theorem and the cone theorem for log canonical pairs. We note that the rationality theorem directly implies the essential part of the cone theorem and that we do not need the theory of quasi-log varieties for the proof of the rationality theorem. In the ﬁnal section, Section 6, we explain some related topics. Acknowledgments. I would like to thank Takeshi Abe for his valuable comments. I would also like to thank Professor Gerard van der Geer and the referee for valuable suggestions and comments. 1.1. Notation and Conventions. We will work over the complex number ﬁeld C throughout this paper. But we note that by using the Lefschetz principle, we can extend everything to the case where the base ﬁeld is an algebraically closed ﬁeld of characteristic zero. We will use the following notation and the notation in [KM] freely. %r Notation. (i) For a Q-Weil divisor D = j=1 dj Dj such that Dj is a prime divisor %r for every j and Di K= Dj for i K= j, we deﬁne the round-up !D" = j=1 !dj "Dj %r (resp. the round-down %D& = j=1 %dj &Dj ), where for every rational number x, !x" (resp. %x&) is the integer deﬁned by x ≤ !x" < x + 1 (resp. x − 1 < %x& ≤ x). The fractional part {D} of D denotes D − %D&. We deﬁne D=1 =

M dj =1

Dj , and D<1 =

M

dj Dj .

dj <1

We call D a boundary (resp. subboundary) Q-divisor if 0 ≤ dj ≤ 1 (resp. dj ≤ 1) for all j. Note that Q-linear equivalence of two Q-divisors B1 and B2 is denoted by B1 ∼Q B2 . (ii) For a proper birational morphism f : X → Y , the exceptional locus Exc(f ) ⊂ X is the locus where f is not an isomorphism. (iii) Let X be a normal variety and B an eﬀective Q-divisor on X such that KX + B is Q-Cartier. Let f : Y → X be a resolution such that Exc(f ) ∪ f∗−1 B has −1 a simple normal crossing support, % where f∗ B is the strict transform of B on Y . ∗ We write KY = f (KX + B) + i ai Ei and a(Ei , X, B) = ai . We say that (X, B) is lc if and only if ai ≥ −1 for all i. Here, lc is an abbreviation of log canonical. Note that the discrepancy a(E, X, B) ∈ Q can be deﬁned for every prime divisor E over X. Let (X, B) be an lc pair. If E is a prime divisor over X such that a(E, X, B) = −1, then the center cX (E) is called an lc center of (X, B).

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291

2. Vanishing and torsion-free theorems In this section, we quickly review Ambro’s formulation of torsion-free and vanishing theorems in a simpliﬁed form (see [F7, Chapter 2]). First, we ﬁx the notation and the conventions to state theorems. 2.1 (Global embedded simple normal crossing pairs). Let Y be a simple normal crossing divisor on a smooth variety M and D a Q-divisor on M . Assume that Supp (D + Y ) is simple normal crossing and that D and Y have no common irreducible components. We put BY = D|Y and consider the pair (Y, BY ). We call (Y, BY ) a global embedded simple normal crossing pair. Let ν : Y ν → Y be the normalization. We put KY ν + Θ = ν ∗ (KY + BY ). A stratum of (Y, BY ) is an irreducible component of Y or the image of some lc center of (Y ν , Θ=1 ). When Y is smooth and BY is a Q-divisor on Y such that Supp BY is simple normal crossing, we put M = Y × A1 and D = BY × A1 . Then (Y, BY ) @ (Y × {0}, BY × {0}) satisﬁes the above conditions, that is, we can consider (Y, BY ) to be a global embedded simple normal crossing pair. Theorem 2.2 is a special case of the main result in [F7, Chapter 2]. It will play crucial roles in the following sections. Theorem 2.2 (Torsion-freeness and vanishing theorem). Let (Y, BY ) be as above. Assume that BY is a boundary Q-divisor. Let f : Y → X be a proper morphism and L a Cartier divisor on Y . (1) Assume that L−(KY +BY ) is f -semi-ample. Then, for every integer q, every non-zero local section of Rq f∗ OY (L) contains in its support the f -image of some stratum of (Y, BY ). (2) Assume that X is projective and L − (KY + BY ) ∼Q f ∗ H for some ample Q-Cartier Q-divisor H on X. Then H p (X, Rq f∗ OY (L)) = 0 for every p > 0 and q ≥ 0. Remark 2.3. It is obvious that the statement of Theorem 2.2 (1) is equivalent to the following one. (1D ) Assume that L − (KY + BY ) is f -semi-ample. Then, for every integer q, every associated prime of Rq f∗ OY (L) is the generic point of the f -image of some stratum of (Y, BY ). The above theorem follows from the next theorem. Theorem 2.4 (Injectivity theorem). Let (Y, BY ) be as above. Assume that Y is proper and BY is a boundary Q-divisor. Let D be an eﬀective Cartier divisor whose support is contained in Supp {BY }. Assume that L ∼Q KY + BY . Then the homomorphism H q (Y, OY (L)) → H q (Y, OY (L + D)), which is induced by the natural inclusion OY → OY (D), is injective for every q. For the proof, which depends on the theory of mixed Hodge structures, we recommend the reader to see [F7, Chapter 2]. It is because [A, Section 3] seems to be inaccessible.

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Osamu Fujino

2.1. Idea of the proof. We prove a very special case of Theorem 2.4. This subsection is independent of the other sections. So, the reader can skip it. We adopt Koll´ ar’s principle (cf. [KM, Principle 2.46]) here instead of using the arguments by Esnault–Viehweg. We closely follow [KM, 2.4 The Kodaira Vanishing Theorem]. We note that [F6] may help the reader to understand Theorem 2.2. In [F6], we give a short and almost self-contained proof of Theorem 2.2 for the case when Y is smooth. First, we recall the following Hodge theoretic results. Note that we compute the cohomology groups in the complex analytic setting throughout this subsection. Theorem 2.5. Let V be a smooth projective variety and Σ a simple normal crossing divisor on V . Let ι : V \ Σ → V be the natural open immersion. Then the inclusion ι! CV \Σ ⊂ OV (−Σ) induces surjections Hci (V \ Σ, C) = H i (V, ι! CV \Σ ) → H i (V, OV (−Σ)) for all i. We note that ι! CV \Σ is quasi-isomorphic to the complex Ω•V (log Σ) ⊗ OV (−Σ) and the Hodge to de Rham spectral sequence E1p,q = H q (V, ΩpV (log Σ) ⊗ OV (−Σ)) =⇒ Hcp+q (V \ Σ, C) degenerates at the E1 -term. See, for example, [E, I.3.], [F7, Section 2.4], or Remark 2.6 below. Theorem 2.5 is a direct consequence of this E1 -degeneration. Remark 2.6. We put n = dim V . By Poincar´e duality, we have H 2n−(p+q) (V \ Σ, C) @ Hcp+q (V \ Σ, C)∗ . On the other hand, by Serre duality, we see that H n−q (V, Ωn−p (log Σ)) @ H q (V, ΩpV (log Σ) ⊗ OV (−Σ))∗ . V Therefore, the above E1 -degeneration easily follows from the well-known E1 -degeneration of D n−p,n−q E1

= H n−q (V, Ωn−p (log Σ)) =⇒ H 2n−(p+q) (V \ Σ, C). V

The next theorem is a special case of Theorem 2.4 if we put Y = X, L = d KX + S + M , and BY = S + m D. Theorem 2.7. Let X be a smooth projective variety and S a simple normal crossing divisor on X. Let M be a Cartier divisor on X. Assume that there exists a smooth divisor D on X such that dD ∼ mM for some relatively prime positive integers d and m with d < m, D and S have no common irreducible components, and D + S is a simple normal crossing divisor on X. Then the homomorphism H i (X, OX (KX + S + M )) → H i (X, OX (KX + S + M + bD)) induced by the natural inclusion OX → OX (bD) is injective for every positive integer b and every i ≥ 0.

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Proof. We take the usual normalization of the m-fold cyclic cover π : Y → X ramiﬁed along the divisor D and deﬁned by dD ∼ mM . We put T = π ∗ S. Then Y is smooth and T is simple normal crossing on Y . Let ι : Y \ T → Y be the natural open immersion. Then the inclusion ι! CY \T ⊂ OY (−T ) induces the following surjections H i (Y, ι! CY \T ) → H i (Y, OY (−T )) for all i by Theorem 2.5. Since the ﬁbers of π are zero-dimensional, there are no higher direct image sheaves, and H i (X, π∗ ι! CY \T ) → H i (X, π∗ OY (−T )) is surjective for every i ≥ 0. The Z/mZ-action gives eigensheaf decompositions π∗ ι! CY \T =

m−1 .

Gk

k=0

and π∗ OY (−T ) =

m−1 . k=0

such that

OX (−S − kM + \ kd m LD)

Gk ⊂ OX (−S − kM + \ kd m LD)

for 0 ≤ k ≤ m − 1. By taking the k = 1 summand, we have the surjections H i (X, G1 ) → H i (X, OX (−S − M )) for all i. It is easy to see that G1 is a subsheaf of OX (−S − M − bD) for every b ≥ 0. See, for example, [KM, Corollary 2.54, Lemma 2.55]. Therefore, H i (X, OX (−S − M − bD)) → H i (X, OX (−S − M )) is surjective for every i (cf. [KM, Corollary 2.56]). By Serre duality, we have the desired injections. By Theorem 2.7, we can easily obtain a very special case of Theorem 2.2 (2). We omit the proof because it is routine work. See, for example, [F1, Section 2.2]. Theorem 2.8. Let f : X → Y be a morphism from a smooth projective variety X onto a projective variety Y . Let S be a simple normal crossing divisor on X and L an ample Cartier divisor on Y . Then H i (Y, Rj f∗ OX (KX + S) ⊗ OY (L)) = 0 for i > 0 and j ≥ 0. As a corollary, we obtain a generalization of the Kodaira vanishing theorem (cf. [F6, Theorem 4.4]).

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Corollary 2.9 (Kodaira vanishing theorem for log canonical varieties). Let Y be a projective variety with only log canonical singularities and L an ample Cartier divisor on Y . Then H i (Y, OY (KY + L)) = 0 for i > 0. Proof. Let f : X → Y be a resolution such that S = Exc(f ) is a simple normal crossing divisor. Then f∗ OX (KX + S) @ OY (KY ). Therefore, we have the desired vanishing theorem by Theorem 2.8. We close this subsection with Sommese’s example. For the details and other examples, see [F7, Section 2.8]. Example 2.10. We consider π : Y = PP1 (OP1 ⊕ OP1 (1)⊕3 ) → P1 . Let M denote the tautological line bundle of π : Y → P1 . We take a general member X of |(M ⊗ π ∗ OP1 (−1))⊗4 |. Then X is a normal Gorenstein projective threefold. Note that X is not lc. We put OY (L) = M ⊗ π ∗ OP1 (1). Then L is an ample Cartier divisor on Y . We can check that H 1 (X, OX (KX + L)) = C. Thus, the Kodaira vanishing theorem does not necessarily hold for non-lc varieties.

3. Adjunction for qlc varieties To prove the base point free theorem for log canonical pairs following Ambro’s idea, it is better to introduce the notion of qlc varieties. For the details, see [F7, Section 3.2]. Deﬁnition 3.1 (Qlc varieties). A qlc variety is a variety X with a Q-Cartier Qdivisor ω, and a ﬁnite collection {C} of reduced and irreducible subvarieties of X such that there is a proper morphism f : (Y, BY ) → X from a global embedded simple normal crossing pair as in 2.1 satisfying the following properties: (1) f ∗ ω ∼Q KY + BY such that BY is a subboundary Q-divisor. (2) There is an isomorphism OX @ f∗ OY (!−(BY<1 )"). (3) The collection of subvarieties {C} coincides with the image of the (Y, BY )strata. We use the following terminology. The subvarieties C are the qlc centers of X, and f : (Y, BY ) → X is a quasi-log resolution of X. We sometimes simply say that [X, ω] is a qlc pair, or the pair [X, ω] is qlc. Remark 3.2. By condition (2), we have an isomorphism OX @ f∗ OY . In particular, f is a surjective morphism with connected ﬁbers and X is semi-normal. Proposition 3.3. Let (X, B) be an lc pair. Then [X, KX + B] is a qlc pair.

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Proof. Let f : Y → X be a resolution such that KY + BY = f ∗ (KX + B) and Supp BY is a simple normal crossing divisor. Then OX @ f∗ OY (!−(BY<1 )") because !−(BY<1 )" is eﬀective and f -exceptional. We note that a qlc center C is X itself or an lc center of (X, B). We start with an easy lemma. Lemma 3.4. Let f : Z → Y be a proper birational morphism between smooth varieties and BY a subboundary Q-divisor on Y such that Supp BY is simple normal crossing. Assume that KZ + BZ = f ∗ (KY + BY ) and that Supp BZ is simple normal crossing. Then we have f∗ OZ (!−(BZ<1 )") @ OY (!−(BY<1 )"). Proof. From KZ + BZ = f ∗ (KY + BY ), we obtain KZ = f ∗ (KY + BY=1 + {BY }) + f ∗ (%BY<1 &) − %BZ<1 & − BZ=1 − {BZ }. If a(ν, Y, BY=1 + {BY }) = −1 for a prime divisor ν over Y , then we can check that a(ν, Y, BY ) = −1 by using [KM, Lemma 2.45]. Since f ∗ (%BY<1 &) − %BZ<1 & is Cartier, we can easily see that f ∗ (%BY<1 &) = %BZ<1 & + E, where E is an eﬀective f -exceptional divisor. Thus, we obtain f∗ OZ (!−(BZ<1 )") @ OY (!−(BY<1 )"). This completes the proof. The following lemma is very important in the study of qlc pairs. Lemma 3.5. We use the same notation and assumptions as in Lemma 3.4. Let S be a simple normal crossing divisor on Y such that S ⊂ Supp BY=1 . Let T be the union of the irreducible components of BZ=1 that are mapped into S by f . Assume that Supp f∗−1 BY ∪ Exc(f ) is simple normal crossing on Z. Then we have f∗ OT (!−(BT<1 )") @ OS (!−(BS<1 )"), where (KZ + BZ )|T = KT + BT and (KY + BY )|S = KS + BS . Proof. We use the same notation as in the proof of Lemma 3.4. We consider the short exact sequence 0 → OZ (!−(BZ<1 )" − T ) → OZ (!−(BZ<1 )") → OT (!−(BT<1 )") → 0. Since T = f ∗ S − F , where F is an eﬀective f -exceptional divisor, we obtain !−(BZ<1 )" − T = f ∗ (!−(BY<1 )" − S) + E + F.

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Here, we used f ∗ (%BY<1 &) = %BZ<1 & + E in the proof of Lemma 3.4. Therefore, f∗ OZ (!−(BZ<1 )" − T ) @ OY (!−(BY<1 )" − S) ⊗ f∗ OZ (E + F ) @ OY (!−(BY<1 )" − S). It is because E and F are eﬀective and f -exceptional. We note that (!−(BZ<1 )" − T ) − (KZ + {BZ } + (BZ=1 − T )) = −f ∗ (KY + BY ). Therefore, every local section of R1 f∗ OZ (!−(BZ<1 )" − T ) contains in its support the f -image of some stratum of (Z, {BZ } + BZ=1 − T ) by Theorem 2.2 (1). Claim. No strata of (Z, {BZ } + BZ=1 − T ) are mapped into S by f . Proof of Claim. Assume that there is a stratum C of (Z, {BZ } + BZ=1 − T ) such that f (C) ⊂ S. Note that Supp f ∗ S ⊂ Supp f∗−1 BY ∪ Exc(f ) and Supp BZ=1 ⊂ Supp f∗−1 BY ∪ Exc(f ). Since C is also a stratum of (Z, BZ=1 ) and C ⊂ Supp f ∗ S, there exists an irreducible component G of BZ=1 such that C ⊂ G ⊂ Supp f ∗ S. Therefore, by the deﬁnition of T , G is an irreducible component of T because f (G) ⊂ S and G is an irreducible component of BZ=1 . So, C is not a stratum of (Z, {BZ } + BZ=1 − T ). This is a contradiction. On the other hand, f (T ) ⊂ S. Therefore, f∗ OT (!−(BT<1 )") → R1 f∗ OZ (!−(BZ<1 )" − T ) is the zero map by the above claim. Thus, we obtain f∗ OT (!−(BT<1 )") @ OS (!−(BS<1 )") by the following commutative diagram. 0

! OY (!−(B <1 )" − S) Y

0

# ! f∗ OZ (!−(B <1 )" − T ) Z

<

! OY (!−(B <1 )") Y <

# ! f∗ OZ (!−(B <1 )") Z

! OS (!−(B <1 )") S

!0

# ! f∗ OT (!−(B <1 )") T

!0

This completes the proof. The following theorem (cf. [A, Theorem 4.4]) is one of the key results for the theory of qlc varieties. It is a consequence of Theorem 2.2. See also Theorem 5.2 below. Theorem 3.6 (Adjunction and vanishing theorem). Let [X, ω] be a qlc pair and X D a union of some qlc centers of [X, ω]. (i) Then [X D , ω D ] is a qlc pair, where ω D = ω|X " . Moreover, the qlc centers of [X D , ω D ] are exactly the qlc centers of [X, ω] that are included in X D .

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(ii) Assume that X is projective. Let L be a Cartier divisor on X such that L − ω is ample. Then H q (X, OX (L)) = 0 and H q (X, IX " ⊗ OX (L)) = 0 for q > 0, where IX " is the deﬁning ideal sheaf of X D on X. Note that H q (X D , OX " (L)) = 0 for every q > 0 because [X D , ω D ] is a qlc pair by (i) and L|X " − ω D is ample. Proof. (i) Let f : (Y, BY ) → X be a quasi-log resolution. Let M be the ambient space of Y and D a subboundary Q-divisor on M such that BY = D|Y . By taking blow-ups of M , we can assume that the union of all strata of (Y, BY ) mapped into X D , which is denoted by Y D , is a union of irreducible components of Y (cf. Lemma 3.5). We put Y DD = Y − Y D . We deﬁne (KY + BY )|Y " = KY " + BY " and consider f : (Y D , BY " ) → X D . We claim that [X D , ω D ] is a qlc pair, where ω D = ω|X " , and f : (Y D , BY " ) → X D is a quasi-log resolution. From the deﬁnition, BY " is a subboundary and f ∗ ω D ∼Q KY " +BY " on Y D . We consider the following short exact sequence 0 → OY "" (−Y D ) → OY → OY " → 0. We put A = !−(BY<1 )". Then we have 0 → OY "" (A − Y D ) → OY (A) → OY " (A) → 0. Applying f∗ , we obtain 0 → f∗ OY "" (A − Y D ) → OX → f∗ OY " (A) → R1 f∗ OY "" (A − Y D ) → · · · . The support of every non-zero local section of R1 f∗ OY "" (A − Y D ) can not be contained in f (Y D ) = X D by Theorem 2.2 (1). We note that −f ∗ ω ∼Q (A − Y D )|Y "" − (KY "" + {BY "" } + BY=1"" − Y D |Y "" ) on Y DD , where (KY + BY )|Y "" = KY "" + BY "" , and that Y D |Y "" is contained in BY=1"" . Therefore, f∗ OY " (A) → R1 f∗ OY "" (A − Y D ) is the zero map. We note that the surjection OX → f∗ OY " (A) decomposes as OX → OX " → f∗ OY " → f∗ OY " (A) since f (Y D ) = X D . Therefore, we obtain OX " @ f∗ OY " (A) = f∗ OY " (!−(BY<1" )"). Thus, we see that f∗ OY "" (A − Y D ) @ IX " , the deﬁning ideal sheaf of X D on X. The statement for qlc centers is obvious by the construction of the quasi-log resolution. So, we obtain (i). (ii) Let f : (Y, BY ) → X be a quasi-log resolution as in the proof of (i). Apply Theorem 2.2 (2). Then we obtain H q (X, OX (L)) = 0 for every q > 0 because f ∗ (L − ω) ∼Q f ∗ L − (KY + BY ) = f ∗ L + !−(BY<1 )" − (KY + {BY } + BY=1 )

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and f∗ OY (f ∗ L + !−(BY<1 )") @ OX (L). We consider f : Y DD → X. We put (KY + BY )|Y "" = KY "" + BY "" . Then f ∗ (L − ω) ∼Q (f ∗ L − (KY + BY ))|Y "" = (f ∗ L + A − Y D )|Y "" − (KY "" + {BY "" } + BY=1"" − Y D |Y "" ) on Y DD . Note that Y D |Y "" is contained in BY=1"" . Therefore, we obtain H q (X, f∗ OY "" (A − Y D ) ⊗ OX (L)) = 0 for every q > 0 by Theorem 2.2 (2). Thus this completes the proof by the isomorphism f∗ OY "" (A − Y D ) @ IX " obtained in the proof of (i). Corollary 3.7. Let [X, ω] be a qlc pair and X D an irreducible component of X. Then [X D , ω D ], where ω D = ω|X " , is a qlc pair. Proof. By Deﬁnition 3.1 and Remark 3.2, X D is a qlc center of [X, ω]. Therefore, by Theorem 3.6 (i), [X D , ω D ] is a qlc pair. We use the next deﬁnition in Section 4. Deﬁnition 3.8. Let [X, ω] be a qlc pair. Let X D be the union of qlc centers of X that are not any irreducible components of X. Then X D with ω D = ω|X " is a qlc variety by Theorem 3.6 (i). We denote it by Nqklt(X, ω). We close this section with the following very useful lemma, which seems to be indispensable for the proof of the base point free theorem in Section 4. Lemma 3.9. Let f : (Y, BY ) → X be a quasi-log resolution of a qlc pair [X, ω]. Let E be a Cartier divisor on X such that Supp E contains no qlc centers of [X, ω]. By blowing up M , the ambient space of Y , inside Supp f ∗ E, we can assume that (Y, BY + f ∗ E) is a global embedded simple normal crossing pair. Proof. First, we take a blow-up of M along f ∗ E and apply Hironaka’s resolution theorem to M . Then we can assume that there exists a Cartier divisor F on M such that Supp (F ∩ Y ) = Supp f ∗ E. Next, we apply Szab´o’s resolution lemma to Supp (D + Y + F ) on M . Thus, we obtain the desired properties by Lemma 3.5.

4. Base point free theorem The next theorem is the main theorem of this section. It is a special case of [A, Theorem 5.1]. This formulation is indispensable for the inductive treatment of log canonical pairs in the framework of the theory of quasi-log varieties. For the details, see [F7, Section 3.2.2]. Theorem 4.1. Let [X, ω] be a projective qlc pair and L a nef Cartier divisor on X. Assume that qL − ω is ample for some q > 0. Then OX (mL) is generated by global sections for every m D 0, that is, there exists a positive number m0 such that OX (mL) is generated by global sections for every m ≥ m0 .

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Proof. First, we note that the statement is obvious when dim X = 0. Claim 1. We can assume that X is irreducible. Let X D be an irreducible component of X. Then X D with ω D = ω|X " has a natural qlc structure induced by [X, ω] by adjunction (see Corollary 3.7). By the vanishing theorem (see Theorem 3.6 (ii)), we have H 1 (X, IX " ⊗ OX (mL)) = 0 for all m ≥ q. We consider the following commutative diagram. H 0 (X, OX (mL)) ⊗ OX

α

# OX (mL)

! H 0 (X D , OX " (mL)) ⊗ OX "

!0

# ! OX " (mL)

!0

Since α is surjective for m ≥ q, we can assume that X is irreducible when we prove this theorem. Claim 2. For every m D 0, OX (mL) is generated by global sections on an open neighborhood of Nqklt(X, ω). We put X D = Nqklt(X, ω). Then [X D , ω D ], where ω D = ω|X " , is a qlc pair by adjunction (see Deﬁnition 3.8 and Theorem 3.6 (i)). By induction on the dimension, OX " (mL) is generated by global sections for every m D 0. By the following commutative diagram: H 0 (X, OX (mL)) ⊗ OX # OX (mL)

α

! H 0 (X D , OX " (mL)) ⊗ OX "

!0

# ! OX " (mL)

! 0,

we know that, for every m D 0, OX (mL) is generated by global sections on an open neighborhood of X D . Claim 3. For every m D 0, OX (mL) is generated by global sections on a nonempty Zariski open set. By Claim 2, we can assume that Nqklt(X, ω) is empty. If L is numerically trivial, then H 0 (X, OX (L)) = H 0 (X, OX (−L)) = C. It is because h0 (X, OX (±L)) = χ(X, OX (±L)) = χ(X, OX ) = 1 by Theorem 3.6 (ii) and [Kl, Chapter II §2 Theorem 1]. Therefore, OX (L) is trivial. So, we can assume that L is not numerically trivial. Let f : (Y, BY ) → X be a quasi-log resolution. Let x ∈ X be a general smooth point. Then we can take a Q-divisor D such that multx D > dim X and D ∼Q (q + r)L − ω for some r > 0 (see [KM, 3.5 Step 2]). By blowing up M , we can assume that (Y, BY + f ∗ D) is a global embedded simple normal crossing pair by Lemma 3.9. We note that every stratum of (Y, BY ) is mapped onto X by the assumption. By the construction of D, we can ﬁnd a positive rational number c < 1 such that BY + cf ∗ D is a subboundary and some stratum of (Y, BY + cf ∗ D) does not dominate X. Note that f∗ OY (!−(BY<1 )") @ OX . Then

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the pair [X, ω+cD] is qlc and f : (Y, BY +cf ∗ D) → X is a quasi-log resolution. We note that q D L − (ω + cD) is ample since c < 1, where q D = q + cr. By construction, Nqklt(X, ω + cD) is non-empty. Therefore, by applying Claim 2 to [X, ω + cD], for every m D 0, OX (mL) is generated by global sections on an open neighborhood of Nqklt(X, ω + cD). So, we obtain Claim 3. Let p be a prime number and l a large integer. Then |pl L| = K ∅ by Claim 3 and l |p L| is free on an open neighborhood of Nqklt(X, ω) by Claim 2. Claim 4. If the base locus Bs|pl L| (with reduced scheme structure) is not empty, " then Bs|pl L| is strictly smaller than Bs|pl L| for some lD > l. Let f : (Y, BY ) → X be a quasi-log resolution. We take a general member D ∈ |pl L|. We note that |pl L| is free on an open neighborhood of Nqklt(X, ω). Thus, f ∗ D intersects all strata of (Y, Supp BY ) transversally over X \ Bs|pl L| by Bertini and f ∗ D contains no strata of (Y, BY ). By taking blow-ups of M suitably, we can assume that (Y, BY + f ∗ D) is a global embedded simple normal crossing pair (cf. Lemmas 3.9 and 3.5). We take the maximal positive rational number c such that BY + cf ∗ D is a subboundary. We note that c ≤ 1. Here, we used OX @ f∗ OY (!−(BY<1 )"). Then f : (Y, BY + cf ∗ D) → X is a quasi-log resolution of [X, ω D = ω + cD]. Note that [X, ω D ] has a qlc center C that intersects Bs|pl L| by the construction. By induction on the dimension, OC (mL) is generated by global sections for all m D 0. We can lift the sections of OC (mL) to X for m ≥ q + cpl by Theorem 3.6 (ii). Then we obtain that, for every m D 0, OX (mL) is generated " by global sections on an open neighborhood of C. Therefore, Bs|pl L| is strictly smaller than Bs|pl L| for some lD > l. Claim 5. OX (mL) is generated by global sections for every m D 0. l"

By Claim 4 and noetherian induction, OX (pl L) and OX (pD L) are generated by global sections for large l and lD , where p and pD are prime numbers and p = K pD . So, there exists a positive number m0 such that OX (mL) is generated by global sections for every m ≥ m0 . The next corollary is obvious from Theorem 4.1 and Proposition 3.3. Corollary 4.2 (Base point free theorem for lc pairs). Let (X, B) be a projective lc pair and L a nef Cartier divisor on X. Assume that qL − (KX + B) is ample for some q > 0. Then OX (mL) is generated by global sections for every m D 0. The reader can ﬁnd another proof of Corollary 4.2 in [F8, Section 4]. It does not need the notion of qlc pairs.

5. Cone theorem In this section, we will state the cone theorem for lc pairs (cf. Theorem 5.3). The essential part of the cone theorem follows from the rationality theorem, Theorem 5.1. The rationality theorem is in turn implied by the vanishing theorem for

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lc centers (cf. Theorem 5.2) by the standard argument (for the details, see [F8, Section 5]). Note that Theorem 5.2 is a special case of Theorem 3.6 (ii), but it can be proved much more easily (see, for example, [F6, Theorem 4.1] or [F8, Theorem 2.2]). Note that we do not need the theory of quasi-log varieties in this section. So, we omit the details. 5.1. Rationality theorem. Here, we explain the rationality theorem for log canonical pairs. It implies the essential part of the cone theorem for log canonical pairs. Theorem 5.1 (Rationality theorem). Let (X, B) be a projective lc pair such that a(KX + B) is Cartier for a positive integer a. Let H be an ample Cartier divisor on X. Assume that KX + B is not nef. We put r = max{t ∈ R : H + t(KX + B) is nef }. Then r is a rational number of the form u/v (u, v ∈ Z) where 0 < v ≤ a(dim X +1). As we explained above, Theorem 5.1 can be proved easily by using the following very special case of Theorem 3.6 (ii). Theorem 5.2 (Vanishing theorem for lc centers). Let X be a projective variety and B a boundary Q-divisor on X such that (X, B) is log canonical. Let D be a Cartier divisor on X. Assume that D − (KX + B) is ample. Let C be an lc center of the pair (X, B) with the reduced scheme structure. Then we have H i (X, IC ⊗ OX (D)) = 0, H i (C, OC (D)) = 0 for all i > 0, where IC is the deﬁning ideal sheaf of C on X. In particular, the restriction map H 0 (X, OX (D)) → H 0 (C, OC (D)) is surjective. The reader can ﬁnd the details of the rationality theorem in [F8, Section 5]. 5.2. Cone theorem. Let us state the main theorem of this section. Theorem 5.3 (Cone theorem). Let (X, B) be a projective lc pair. Then we have (i) There are (countably many) rational curves Cj ⊂ X such that 0 < −(KX + B) · Cj ≤ 2 dim X, and M N E(X) = N E(X)(KX +B)≥0 + R≥0 [Cj ]. (ii) For any ε > 0 and ample Q-divisor H, N E(X) = N E(X)(KX +B+εH)≥0 +

M ﬁnite

R≥0 [Cj ].

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(iii) Let F ⊂ N E(X) be a (KX + B)-negative extremal face. Then there is a unique morphism ϕF : X → Z such that (ϕF )∗ OX @ OZ , Z is projective, and an irreducible curve C ⊂ X is mapped to a point by ϕF if and only if [C] ∈ F . The map ϕF is called the contraction of F . (iv) Let F and ϕF be as in (iii). Let L be a line bundle on X such that (L · C) = 0 for every curve C with [C] ∈ F . Then there is a line bundle LZ on Z such that L @ ϕ∗F LZ . Proof. The upper bound 2 dim X and the fact that Cj is a rational curve in (i) can be proved by Kawamata’s argument in [Ka] with the aid of [BCHM]. For the details, see [F7, Section 3.1.3] or [F9, Section 18]. The other statements in (i) and (ii) are formal consequences of the rationality theorem (cf. Theorem 5.1). For the proof, see [KM, Theorem 3.15]. The statements (iii) and (iv) are obvious by Corollary 4.2 and the statements (i) and (ii). See Steps 7 and 9 in [KM, 3.3 The Cone Theorem].

6. Related topics In this paper, we did not prove Theorem 2.2, which is a key result for the theory of quasi-log varieties. For the proof, see [F7, Chapter 2]. The paper [F6] is a gentle introduction to the vanishing and torsion-free theorems. In [F7, Chapters 3, 4], we give a proof of the existence of fourfold lc ﬂips and prove the base point free theorem of Reid–Fukuda type for lc pairs. The base point free theorem for lc pairs is generalized in [F2], where we obtain Koll´ ar’s eﬀective base point free theorem for lc pairs. In [F3], we prove the eﬀective base point free theorem of Angehrn–Siu type for lc pairs. We introduce the notion of non-lc ideal sheaves and prove the restriction theorem in [F4]. It is a generalization of Kawakita’s inversion of adjunction on log canonicity for normal divisors. See also [FST]. In [F5], we prove that the log canonical ring is ﬁnitely generated in dimension four. In [F8], we obtain the fundamental theorems of the log minimal model program for log canonical pairs without using the theory of quasi-log varieties. Our new approach in [F8] seems to be more natural and simpler than Ambro’s theory of quasi-log varieties. In [F9], we go ahead with this new approach. We strongly recommend the reader to see [F8] and [F9]. Finally, in [F10], the minimal model theory for log surfaces is discussed under much weaker assumptions than everybody expected.

7. References [A]

F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220–239; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 214–233.

[BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23, no. 2, 405–468.

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303

[E]

F. Elzein, Mixed Hodge structures, Trans. Amer. Math. Soc. 275 (1983), no. 1, 71–106.

[F1]

O. Fujino, Higher direct images of log canonical divisors, J. Diﬀerential Geom. 66 (2004), no. 3, 453–479.

[F2]

O. Fujino, Eﬀective base point free theorem for log canonical pairs—Koll´ ar type theorem, Tohoku Math. J. 61 (2009), 475–481.

[F3]

O. Fujino, Eﬀective base point free theorem for log canonical pairs II—Angehrn– Siu type theorems—, to appear in Michigan Math. J.

[F4]

O. Fujino, Theory of non-lc ideal sheaves: basic properties, Kyoto Journal of Mathematics, Vol. 50, No. 2 (2010), 225–245.

[F5]

O. Fujino, Finite generation of the log canonical ring in dimension four, to appear in Nagata memorial issue of Kyoto Journal of Mathematics.

[F6]

O. Fujino, On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 8, 95–100.

[F7]

O. Fujino, Introduction to the log minimal model program for log canonical pairs, preprint 2009.

[F8]

O. Fujino, Non-vanishing theorem for log canonical pairs, to appear in Journal of Algebraic Geometry.

[F9]

O. Fujino, Fundamental theorems for the log minimal model program, preprint 2009.

[F10]

O. Fujino, Minimal model theory for log surfaces, preprint 2010.

[FST]

O. Fujino, K. Schwede, and S. Takagi, Supplements to non-lc ideal sheaves, preprint 2010.

[Ka]

Y. Kawamata, On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609–611.

[Kl]

S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344.

[KM]

J. Koll´ ar, S. Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.

Osamu Fujino, Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502 Japan E-mail: [email protected]

On Kawamata’s theorem Osamu Fujino∗

Abstract. We give an alternate proof of the main theorem of Kawamata’s paper: Pluricanonical systems on minimal algebraic varieties. Our proof also works for varieties in class C. We note that our proof is completely diﬀerent from Kawamata’s. 2010 Mathematics Subject Classiﬁcation. Primary 14C20; Secondary 14E30. Keywords. Base point free theorem, Abundance conjecture, Canonical bundle formula.

1. Introduction One of the main purposes of this paper is to cut a chain of troubles caused by [Ka, Theorem 4.3]. We give an alternate proof of the following famous theorem, which we call Kawamata’s theorem in this paper. This theorem is indispensable for the abundance conjecture. Theorem 1.1 (cf. [KMM, Theorem 6-1-11]). Let (X, B) be a klt pair and π : X → S a proper surjective morphism of normal varieties. Assume the following conditions: (a) H is a π-nef Q-Cartier divisor on X, (b) H − (KX + B) is π-nef and π-abundant, and (c) κ(Xη , (aH − (KX + B))η ) ≥ 0 and ν(Xη , (aH − (KX + B))η ) = ν(Xη , (H − (KX + B))η ) for some a ∈ Q with a > 1, where η is the generic point of S. Then H is π-semi-ample. It was ﬁrst proved in [Ka] on the assumption that S is a point. Kawamata’s proof heavily depends on a very technical generalization of Koll´ar’s injectivity theorem on generalized normal crossing varieties (see [Ka, Section 4]). Once we adopt this diﬃcult injectivity theorem, the X-method works and the proof is essentially the same as the one of the Kawamata–Shokurov base point free theorem. Unfortunately, there is an ambiguity in the proof of [Ka, Theorem 4.3] (see [F2, Remark 3.10.3] and 5.1 below). Thus, our proof is the ﬁrst rigorous proof of Kawamata’s theorem. It is completely diﬀerent from Kawamata’s. His proof relies on the theory of mixed Hodge structures for reducible varieties. Our proof grew out from the theory of variation of Hodge structures, especially, Deligne’s canonical extensions ∗ The

author was partially supported by The Sumitomo Foundation, The Inamori Foundation, and by the Grant-in-Aid for Young Scientists (A) 320684001 from JSPS.

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of Hodge bundles. We note that our method saves Kawamata’s theorem but does not recover the results in [Ka, Section 4]. They are completely generalized in [F4, Chapter 2] for embedded simple normal crossing pairs. However, [F4] does not recover [Ka, Theorem 4.3]. Compare the arguments in [F4, Chapter 2] with Kawamata’s ones. The reader can ﬁnd a slight generalization of Kawamata’s theorem and some other applications of our methods in [F3], [F5], and [FG]. We summarize the contents of this paper. In Section 2, we will give an alternate proof of Kawamata’s theorem. By using Ambro’s formula, we will reduce Kawamata’s theorem to a reformulated version of the Kawamata–Shokurov base point free theorem. Section 3 is an appendix, where we will quickly review Ambro’s formula for the reader’s convenience. In Section 4, we will prove Kawamata’s theorem for varieties in class C, which is [N2, Theorem 5.5]. We separate this section from Section 2 in order not to make needless confusion. In the ﬁnal section, Section 5, we will make some comments on topics related to Kawamata’s theorem for the coming generation. This paper was ﬁrst circulated as “A remark on the base point free theorem” on 28 August, 2005 (arXiv:math/0508554v1). Acknowledgments. The author would like to thank Professors Yujiro Kawamata and Noboru Nakayama for answering his questions. He thanks Professors Daisuke Matsushita and Shigefumi Mori for encouraging him during the preparation of this paper. We will work over an algebraically closed ﬁeld k of characteristic zero throughout this paper. We adopt the language of b-divisors and use the standard notation of the log minimal model program. See, for example, [C].

2. Proof of Kawamata’s theorem The following theorem is a reformulation of the Kawamata–Shokurov base point free theorem. The original proof works without any changes (cf. [KMM, Theorem 3-1-1]). Theorem 2.1 (Base point free theorem). Let (X, B) be a sub klt pair, let π : X → S be a proper surjective morphism of normal varieties, and D a π-nef Cartier divisor on X. Assume the following conditions: (1) rD − (KX + B) is nef and big over S for some positive integer r, and (2) π∗ OX (!A(X, B)" + jD) ⊆ π∗ OX (jD) for every positive integer j, where A(X, B) is the discrepancy Q-b-divisor and D is the Cartier closure of D (see [C, Example 2.3.12 (1) (3)]). Then mD is π-generated for m D 0, that is, there exists a positive integer m0 such that for every m ≥ m0 the natural homomorphism π ∗ π∗ OX (mD) → OX (mD) is surjective.

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Before the proof of Theorem 1.1, let us recall the deﬁnition of abundant divisors, which are called good divisors in [Ka]. See [KMM, §6-1]. Deﬁnition 2.2 (Abundant divisor). Let X be a complete normal variety and D a Q-Cartier nef divisor on X. We deﬁne the numerical Iitaka dimension to be ν(X, D) = max{e; De K≡ 0}. "

This means that De · S = 0 for any eD -dimensional subvarieties S of X with eD > e and there exists an e-dimensional subvariety T of X such that De ·T > 0. Then it is easy to see that κ(X, D) ≤ ν(X, D), where κ(X, D) denotes Iitaka’s D-dimension. A nef Q-divisor D is said to be abundant if the equality κ(X, D) = ν(X, D) holds. Let π : X → S be a proper surjective morphism of normal varieties and D a QCartier divisor on X. Then D is said to be π-abundant if D|Xη is abundant, where Xη is the generic ﬁber of π. Proof of Theorem 1.1. If H − (KX + B) is π-big, then the statement follows from the original Kawamata–Shokurov base point free theorem. Thus, from now on, we assume that H − (KX + B) is not π-big. Then there exists a diagram f

Y −−−−→   µN

Z  ϕ N

X −−−−→ S π

which satisﬁes the following conditions (see [KMM, Proposition 6-1-3 and Remark 6-1-4] or [N1, Lemma 6]): (i) µ, f and ϕ are projective morphisms, (ii) Y and Z are non-singular varieties, (iii) µ is a birational morphism and f is a surjective morphism having connected ﬁbers, (iv) there exists a ϕ-nef and ϕ-big Q-divisor M0 on Z such that µ∗ (H − (KX + B)) ∼Q f ∗ M0 , and (v) there is a ϕ-nef Q-divisor D on Z such that µ∗ H ∼Q f ∗ D. Note that f : Y → Z is the Iitaka ﬁbration with respect to H − (KX + B) over S. We put KY + BY = µ∗ (KX + B) and HY = µ∗ H. We note that (Y, BY ) is not necessarily klt but sub klt. Thus, we have HY −(KY +BY ) ∼Q f ∗ M0 (resp. HY ∼Q f ∗ D), where M0 (resp. D) is a ϕ-nef and ϕ-big (resp. ϕ-nef) Q-divisor as we saw in (iv) and (v). Furthermore, we can assume that D and H are Cartier divisors and HY ∼ f ∗ D by replacing D and H by suﬃciently divisible multiples. If necessary, we modify Y and Z birationally and can assume the following conditions:

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(1) KY + BY ∼Q f ∗ (KZ + BZ + M ), where BZ is the discriminant Q-divisor of (Y, BY ) on Z and M is the moduli Q-divisor on Z, (2) (Z, BZ ) is a sub klt pair, (3) M is a ϕ-nef Q-divisor on Z, (4) ϕ∗ OZ (!A(Z, BZ )" + jD) ⊆ ϕ∗ OZ (jD) for every positive integer j, and (5) D − (KZ + BZ ) is ϕ-nef and ϕ-big. Indeed, let P ⊂ Z be a prime divisor. Let aP be the largest real number t such that (Y, BY + tf ∗ P ) is sub lc over the generic point of P . It is obvious that aP = 1 for all but ﬁnitely many prime divisors P of Z. We note that aP is a positive rational number for any P . The discriminant Q-divisor on Z is deﬁned by the following formula M (1 − aP )P. BZ = P

We note that %BZ & ≤ 0. By properties (iv) and (v), we can write KY + BY ∼Q f ∗ (M1 ) for a Q-Cartier divisor M1 on Z. We deﬁne M = M1 − (KZ + BZ ) and call it the moduli Q-divisor on Z, where BZ is the discriminant Q-divisor deﬁned above. Note that M is called the log-semistable part in [FM, Section 4]. So, the condition (1) obviously holds by the deﬁnitions of the discriminant Q-divisor BZ and the moduli Q-divisor M . If we take birational modiﬁcations of Y and Z suitably, we have that M is ϕ-nef and (Z, BZ ) is sub klt. Thus we obtain (2) and (3). For the details, see [A1, Theorems 0.2 and 2.7] or Theorem 3.2 below. We note the following lemma (cf. [A1, Lemma 6.2]), which we need to apply [A1, Theorems 0.2 and 2.7] or Theorem 3.2 to f : Y → Z (see the condition (2) in 3.1). Lemma 2.3. We have rank f∗ OY (!A(Y, BY )") = 1. Proof of Lemma 2.3. Since OZ @ f∗ OY ⊆ f∗ OY (!A(Y, BY )"), we know rank f∗ OY (!A(Y, BY )") ≥ 1. Without loss of generality, we can shrink S and assume that S is aﬃne. Let A be a ϕ-very ample divisor such that f∗ OY (!A(Y, BY )") ⊗ OZ (A) is ϕ-generated. Since M0 is a ϕ-big Q-divisor on Z, we have OZ (A) ⊂ OZ (mM0 ) for a suﬃciently divisible positive integer m. We note that π∗ µ∗ OY (!A(Y, BY )" + f ∗ (mM0 )) @ π∗ µ∗ OY (f ∗ (mM0 )), where f ∗ (mM0 ) is the Cartier closure of f ∗ (mM0 ) (see [C, Example 2.3.12 (1)]). It is because µ∗ (H − (KX + B)) = HY − (KY + BY ) ∼Q f ∗ M0 . Therefore, ϕ∗ (f∗ OY (!A(Y, BY )") ⊗ OZ (A)) ⊆ ϕ∗ (f∗ OY (!A(Y, BY )") ⊗ OZ (mM0 )) @ ϕ∗ OZ (mM0 ). So, we see that rank f∗ OY (!A(Y, BY )") ≤ 1. This completes the proof.

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We know the following lemma by Lemma 9.2.2 and Proposition 9.2.3 in [A2] (see also Theorem 3.2 (a) below). Lemma 2.4. We have OZ (!A(Z, BZ )" + jD) ⊆ f∗ OY (!A(Y, BY )" + jHY ) for every integer j. Pushing forward by ϕ, we obtain that ϕ∗ OZ (!A(Z, BZ )" + jD) ⊆ ϕ∗ f∗ OY (!A(Y, BY )" + jHY ) @ π∗ µ∗ OY (!A(Y, BY )" + jHY ) @ π∗ OX (!A(X, B)" + jH) @ π∗ OX (jH) @ π∗ µ∗ OY (jHY ) @ ϕ∗ f∗ OY (jHY ) @ ϕ∗ OZ (jD) for every integer j. Thus, we have (4). The relation HY − (KY + BY ) ∼Q f ∗ (D − (KZ + BZ + M )) implies that D − (KZ + BZ + M ) is ϕ-nef and ϕ-big. By (3), M is ϕ-nef. Therefore, D − (KZ + BZ ) = D − (KZ + BZ + M ) + M is ϕ-nef and ϕ-big. This is condition (5). Apply Theorem 2.1 to D on (Z, BZ ). Then we obtain that D is ϕ-semi-ample. This implies that H is π-semi-ample. This completes the proof. Theorem 1.1 has the following obvious corollaries. Corollary 2.5. Let (X, B) be a klt pair and π : X → S a proper surjective morphism of normal varieties. Assume that KX + B is π-nef and π-abundant. Then KX + B is π-semi-ample. Corollary 2.6. Let X be a complete normal variety such that KX ∼Q 0. Assume that X has only klt singularities. Let H be a nef and abundant Q-Cartier divisor on X. Then H is semi-ample. We close this section with a useful remark. Remark 2.7 (cf. [F5, Remark 3.5]). Let π : X → S be a proper surjective morphism of normal varieties and D a π-nef and π-abundant Cartier divisor on X. Then we can easily check that . π∗ OX (mD) m≥0

is ﬁnitely generated if and only if D is π-semi-ample. See, for example, [F5, Lemma 3.10].

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Let B be an eﬀective Q-divisor on X such that (X, B) is klt. By [BCHM], we know that . π∗ OX (%m(KX + B)&) m≥0

is ﬁnitely generated. Assume that KX +B is π-nef. By the above observation, we obtain that KX +B is π-semi-ample if and only if KX + B is π-nef and π-abundant. Therefore, we do not need Theorem 1.1 to obtain Corollaries 2.5 and 2.6.

3. Appendix: Quick review of Ambro’s formula In this appendix, we quickly review Ambro’s formula. For the details, see the original paper [A1] or Koll´ ar’s survey article [Ko]. 3.1. Let f : X → Y be a proper surjective morphism of normal varieties and p : Y → S a proper morphism onto a variety S. Assume the following conditions: (1) KX + B is Q-Cartier and (X, B) is sub klt over the generic point of Y , (2) rank f∗ OX (!A(X, B)") = 1, and (3) KX + B ∼Q,f 0. By (3), we can write KX + B ∼Q f ∗ D for some Q-Cartier divisor D on Y . Let BY be the discriminant Q-divisor on Y . For the deﬁnition, see the proof of Theorem 1.1. We put MY = D − (KY + BY ) and call MY the moduli Q-divisor on Y . Then we have KX + B ∼Q f ∗ (KY + BY + MY ). Let σ : Y D → Y be a proper birational morphism from a normal variety Y D . Then we obtain the following commutative diagram: µ X ←−−−− X D    "  fN Nf Y ←−−−− Y D σ

such that (i) µ is a birational morphism from a normal variety X D ; ∗

(ii) if we put KX " +B D = µ∗ (KX +B), then we can write KX " +B D ∼Q f D (KY " + BY " + MY " ), where BY " is the discriminant Q-divisor on Y D associated to f D : X D → Y D. Ambro’s theorem [A1, Theorems 0.2 and 2.7] says Theorem 3.2. If we choose Y D appropriately, then we have the following properties for every proper birational morphism ν : Y DD → Y D from a normal variety Y DD .

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(a) KY " + BY " is Q-Cartier and ν ∗ (KY " + BY " ) = KY "" + BY "" . In particular, A(Y D , BY " )Y "" = −BY "" . (b) The moduli Q-divisor MY " is nef over S and ν ∗ (MY " ) = MY "" . We note that the nefness of the moduli Q-divisor can be proved by using Fujita– Kawamata’s semi-positivity theorem. It is a consequence of the theory of variation of Hodge structures. For details, see, for example, [M, Section 5], [F1, Section 5], or [Ko].

4. Kawamata’s theorem for varieties in class C In this section, we treat Nakayama’s theorem: [N2, Theorem 5.5], which is Kawamata’s theorem for varieties in class C. First, let us recall the deﬁnition of the varieties in class C. Deﬁnition 4.1 (Class C). A compact complex variety in class C is a variety which is dominated by a compact K¨ahler manifold. It is known that X is in class C if and only if X is bimeromorphically equivalent to a compact K¨ ahler manifold. Next, we recall the deﬁnitions of the K¨ ahler cone and the nef line bundles on a compact K¨ ahler manifold. Deﬁnition 4.2 (K¨ahler cone). Let Y be a d-dimensional compact K¨ahler manifold. We deﬁne the K¨ahler cone KC(Y ) of Y to be the set {[ω] ∈ H 1,1 (Y, R); ω is a K¨ ahler form on Y }, where H 1,1 (Y, R) := H 2 (Y, R) ∩ H 1,1 (Y, C). Then KC(Y ) is an open convex cone in H 1,1 (Y, R). KC(Y ) is the closure of KC(Y ) in H 1,1 (Y, R). Deﬁnition 4.3 (cf. [N2, Deﬁnition 2.4]). Let L be a line bundle on a compact K¨ahler manifold Y . L is said to be nef if the real ﬁrst Chern class c1 (L) is contained in KC(Y ). Remark 4.4. For a new numerical characterization of the K¨ ahler cone of a compact K¨ ahler manifold, see [DP, Main Theorem 0.1]. A nef line bundle on a compact K¨ahler manifold can be characterized numerically by [DP, Corollaries 0.3 and 0.4]. Finally, we recall the deﬁnitions of the quasi-nef line bundles, the homological Kodaira dimension, and the big and abundant line bundles, which were introduced in [N2]. Deﬁnition 4.5 (cf. [N2, Deﬁnition 2.6]). Let X be a compact complex variety in class C. A line bundle L on X is called quasi-nef if there exists a bimeromorphic morphism µ : Y → X from a compact K¨ahler manifold Y such that µ∗ L is nef.

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Deﬁnition 4.6 (cf. [N2, Deﬁnition 2.9]). Let L be a quasi-nef line bundle on a complex variety X in class C. Take a bimeromorphic morphism µ : Y → X from a compact K¨ ahler manifold Y such that µ∗ L is nef. Then we deﬁne κhom (L) := max{l ≥ 0; 0 = K c1 (µ∗ L)l ∈ H l,l (Y, R)} and call it the homological Kodaira dimension of L. It is well-deﬁned, because it is independent of the choice of Y . Deﬁnition 4.7 (cf. [N2, Deﬁnition 2.11]). Let L be a line bundle on a compact complex variety X in class C. L is said to be big if κ(X, L) = dim X. If L is quasi-nef and κ(X, L) = κhom (L), then L is called abundant. Now, we state the main theorem of this section. It is nothing but [N2, Theorem 5.5]. The reader can ﬁnd some applications of Theorem 4.8 in [COP]. Theorem 4.8 (cf. [N2, Theorem 5.5]). Let X be a compact normal complex variety in class C, B an eﬀective Q-divisor on X, and H a Q-Cartier divisor on X. Then H is semi-ample under the following conditions: (1) (X, B) is klt, (2) H is quasi-nef, (3) H − (KX + B) is quasi-nef and abundant, and (4) κhom (aH − (KX + B)) = κhom (H − (KX + B)) and κ(X, aH − (KX + B)) ≥ 0 for some a ∈ Q with a > 1. Sketch of the proof. First, we recall Nakamaya’s result. Lemma 4.9 ([N2, Proposition 2.14 and Corollary 2.16]). There exists the following diagram µ f X ←−−−− Y −−−−→ Z, where (a) Y is a compact K¨ ahler manifold and µ is a bimeromorphic morphism, (b) Z is a smooth projective variety, (c) f is surjective and has connected ﬁbers, (d) there exists a nef and big Q-divisor M0 on Z such that µ∗ (H − (KX + B)) ∼Q f ∗ M0 , and (e) there is a nef Q-divisor D on Z such that µ∗ H ∼Q f ∗ D.

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We note that Z is a smooth projective variety. Let f : Y → Z be the proper surjective morphism from a compact K¨ ahler manifold Y to a normal projective variety Z obtained in Lemma 4.9. Let BY be a Q-divisor on Y such that KY + BY = µ∗ (KX + B). Then we have the following properties: (1) KY + BY is Q-Cartier and (Yz , Bz ) is sub klt for general z ∈ Z, where Yz = f −1 (z) and Bz = BY |Yz , (2) rank f∗ OY (!A(Y, BY )") = 1, and (3) KY + BY ∼Q,f 0. We note that (1) is obvious by the deﬁnition of BY , (2) follows from the proof of Lemma 2.3, and (3) is also obvious by Lemma 4.9. Under these conditions (1), (2), and (3), Ambro’s theorem (see [A1, Theorems 0.2 and 2.7] or Theorem 3.2) holds if we use [N3, 3.7. Theorem (4)] in the proof of Ambro’s theorem. Note that it is not diﬃcult to modify the arguments in [A1] for our setting. More explicitly, let σ : Z D → Z be a proper birational morphism from a normal projective variety Z D . If we choose Z D appropriately, then we have the following properties for every proper birational morphism ν : Z DD → Z D from a normal projective variety Z DD . (a) KZ " + BZ " is Q-Cartier and ν ∗ (KZ " + BZ " ) = KZ "" + BZ "" , where BZ " and BZ "" are the discriminant Q-divisors. In particular, A(Z D , BZ " )Z "" = −BZ "" . (b) The moduli Q-divisor MZ " is nef and ν ∗ (MZ " ) = MZ "" . For the details and the notation, see Section 3. By applying Ambro’s theorem to f : Y → Z, the proof of Theorem 1.1 works without any modiﬁcations. We note that Z is a projective variety. Thus, we obtain the semi-ampleness of H.

5. Comments for the coming generation The results in [Ka] had already been used in various papers. We think that almost all the papers only used the main results of [Ka], that is, Theorems 1.1 and 6.1 in [Ka]. Therefore, by this paper, almost all the troubles caused by [Ka, Theorem 4.3] were removed. However, some authors used arguments in [Ka]. We give some comments for the coming generation. 5.1. As we pointed out in [F2, Remark 3.10.3], the proof of [Ka, Theorem 4.3] is not completed (see also [KMM, Theorem 6-1-6]). We recall the trouble in [Ka] here for the reader’s convenience. We use the same notation as in the proof of Theorem 4.3 in [Ka]. By [Ka, Theorem 3.2], DE1p,q → DDE1p,q are zero for all p and q. It does not directly say that H i (X, OX (−!L")) → H i (D, OD (−!L"))

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are zero for all i. So, the proofs of Theorems 4.4, 4.5, 5.1, and 6.1 in [Ka] do not work. It is because everything depends on Theorem 4.3 in [Ka]. Thus, we have no rigorous proofs for [KMM, Theorems 6-1-8, 6-1-9]. In [Ka], there seem to be no troubles except the proof of Theorem 4.3. If someone corrects the proof of [Ka, Theorem 4.3], then the following comments are unnecessary. 5.2. In [N1], Nakayama obtained the relative version of Kawamata’s theorem. The proof given there heavily depends on Kawamata’s original proof. So, it does not work by the trouble in [Ka, Theorem 4.3]. Of course, [N1, Theorem 5] is true by our main theorem: Theorem 1.1. 5.3. Section 5 in [N2] contains the same trouble. It is because it depends on Kawamata’s paper [Ka]. In Section 4, we give a rigorous proof of [N2, Theorem 5.5]. 5.4. In [Fk], Fukuda obtained a slight generalization of Kawamata’s theorem. See [Fk, Proposition 3.3]. In the ﬁnal step of the proof of [Fk, Proposition 3.3], Fukuda used [Ka, Theorem 5.1]. So, Fukuda’s original proof also has some troubles by [Ka, Theorem 4.3]. Fortunately, we can prove a slight generalization of [Fk, Proposition 3.3] in [F3, Section 6].

6. References [A1]

F. Ambro, Shokurov’s boundary property, J. Diﬀerential Geom. 67, no. 2 (2004), 229–255.

[A2]

F. Ambro, Non-klt techniques, in Flips for 3-folds and 4-folds (Alessio Corti, ed.), 163–170, Oxford University Press, 2007.

[BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23, no. 2, 405–468. [COP]

F. Campana, K. Oguiso, and T. Peternell, Non-algebraic Hyperkaehler manifolds, preprint, 2008.

[C]

A. Corti, 3-fold ﬂips after Shokurov, in Flips for 3-folds and 4-folds (Alessio Corti, ed.), 18–48, Oxford University Press, 2007.

[DP]

J-P. Demailly, and M. Paun, Numerical characterization of the K¨ ahler cone of a compact K¨ ahler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247–1274.

[F1]

O. Fujino, Remarks on algebraic ﬁber spaces, J. Math. Kyoto Univ. 45 (2005), no. 4, 683–699.

[F2]

O. Fujino, What is log terminal?, in Flips for 3-folds and 4-folds (Alessio Corti, ed.), 49–62, Oxford University Press, 2007.

[F3]

O. Fujino, Base point free theorems—saturation, b-divisors, and canonical bundle formula—, preprint 2007.

[F4]

O. Fujino, Introduction to the log minimal model program for log canonical pairs, preprint 2008.

On Kawamata’s theorem

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[F5]

O. Fujino, Finite generation of the log canonical ring in dimension four, to appear in Nagata memorial issue of Kyoto Journal of Mathematics.

[FG]

O. Fujino, and Y. Gongyo, On images of weak Fano manifolds, preprint 2010.

[FM]

O. Fujino, and S. Mori, A canonical bundle formula, J. Diﬀerential Geom. 56 (2000), no. 1, 167–188.

[Fk]

S. Fukuda, On numerically eﬀective log canonical divisors, Int. J. Math. Math. Sci. 30 (2002), no. 9, 521–531.

[Ka]

Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588.

[KMM]

Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.

[Ko]

J. Koll´ ar, Kodaira’s canonical bundle formula and adjunction, in Flips for 3folds and 4-folds (Alessio Corti, ed.), 134–162, Oxford University Press, 2007.

[M]

S. Mori, Classiﬁcation of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 269–331, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.

[N1]

N. Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, Topology 25 (1986), no. 2, 237–251.

[N2]

N. Nakayama, The lower semicontinuity of the plurigenera of complex varieties, Algebraic geometry, Sendai, 1985, 551–590, Adv. Stud. Pure Math., 10, NorthHolland, Amsterdam, 1987.

[N3]

N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004.

Osamu Fujino, Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502 Japan E-mail: [email protected]

Remarks on the cone of divisors Yujiro Kawamata

Abstract. We prove two theorems on the locally ﬁnite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of numerical linear systems in order to simplify the argument on the Zariski decompositions. 2010 Mathematics Subject Classiﬁcation. 14E30

1. Introduction In the Minimal Model Program (MMP), a contraction morphism arises from an extremal ray on a cone of curves, a subset of a vector space N1 (X/S) consisting of numerical classes of eﬀective 1-cycles. This is like a homology theory. According to the later development of the MMP, it turned out that the subsets of the dual vector space N 1 (X/S), the vector space of divisors, are more useful. This is like a cohomology theory. In this paper we shall make some remarks on the ﬁniteness properties on subsets of N 1 (X/S). Let f : X → S be a projective morphism between algebraic schemes. Two R-Cartier divisors D1 and D2 , linear combinations of Cartier divisors with real coeﬃcients, on X are said to be numerically equivalent over S, and denoted by D1 ≡S D2 , or simply D1 ≡ D2 , if equalities (D1 · C) = (D2 · C) hold for all curves C on X which are mapped to points on S. The set of all the numerical classes of R-Cartier divisors forms a ﬁnite dimensional real vector space N 1 (X/S). We consider the following inclusions of convex cones inside N 1 (X/S): Amp(X/S) −−−−→ Nef(X/S)     N N Big(X/S) −−−−→ Psef(X/S) where the ample cone Amp(X/S) is the open convex cone generated by the classes of Cartier divisors which are ample over S, the nef cone Nef(X/S) is its closure, the pseudo-eﬀective cone Psef(X/S) is the closed convex cone generated by the classes of eﬀective Cartier divisors, and the big cone Big(X/S) is its interior (cf. [4]). We shall prove two theorems on the locally ﬁnite decompositions of these cones in §4 which are determined respectively by the canonical and minimal models (Theorems 3 and 4). The assertions of the theorems are basically contained in a paper by Shokurov [8]. But we make the statements more precise, especially in

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the second theorem. We considered a partial decomposition of the vector space N 1 (X/S) by nef cones of various birationally equivalent minimal models in [4]. That was a locally ﬁnite decomposition. Shokurov’s idea is to consider the space of boundary divisors themselves instead of their numerical classes, so that the assertions of theorems become not only local ﬁniteness but also global ﬁniteness. We treat log pairs whose boundaries are not necessarily big, but we assume that there exist minimal and canonical models of the pairs (cf. [1]). We include examples at the end of the paper on the whole space N 1 (X/S) where the ﬁniteness holds only locally. We introduce the concept of numerical linear systems in §3 which replaces that of R-linear systems in [1] in order to simplify the argument on the Zariski decompositions. We identify the exceptional divisors of the birational map to a minimal model as the numerical ﬁxed part of the numerical canonical system (Lemma 2). This lemma is used to prove the ﬁnite decomposition theorem according to the choices of minimal models (Theorem 4).

2. Minimal and canonical models The purpose of this section is to ﬁx the notation and the terminology. The MMP deals with pairs (X, B) consisting of normal varieties and R-divisors. Let µ : Y → X be a log resolution of the pair (X, B), i.e., µ is a birational morphism from a smooth variety Y such that the inverse image of B and the exceptional locus of µ as well as their union are normal crossing divisors. We can write µ∗ (KX + B) = % KY + C with irreducible decomposition C = j cj Cj . Then the pair is said to be LC (resp. KLT) if cj ≤ 1 (resp. cj < 1) for all j. This deﬁnition does not depend on the choice of µ. The pair is called DLT if there exists a log resolution µ such that cj ≤ 1 for all j with strict inequalities when codim µ(Cj ) ≥ 2. The cone theorem ([3]) can be stated in the following way: Theorem 1. Let f : X → S be a projective morphism from a DLT pair (X, B). Then the cone of nef divisors Nef(X/S) in N 1 (X/S) looks locally rational polyhedral when observed from the numerical class of KX +B in the following sense: let V be the part of the boundary ∂Nef(X/S) which is visible from the point v0 = [KX +B] V = {v ∈ ∂Nef(X/S) | [v, v0 ] ∩ Nef(X/S) = {v}}, then any compact subset in the relative interior of V consists of ﬁnitely many faces which are deﬁned by linear equations with rational coeﬃcients. Let (X, B) be an LC pair consisting of a normal Q-factorial variety and an R-divisor with a projective morphism f : X → S to a base space. We assume ¯ such that (X, B) ¯ an additional condition that there exists another boundary B is a KLT pair. Then the MMP or the MMP with scaling for the morphism f : (X, B) → S works and preserves the situation in the sense that the cone and contraction theorems hold and the resulting morphism after a divisorial contraction

319 or a ﬂip satisﬁes the same condition. The existence of ﬂips is proved in [2], but the termination of ﬂips in general is not yet. A minimal model for a projective morphism f : (X, B) → S is deﬁned to be a projective morphism g : (Y, C) → S with a birational map α : X ''( Y over S which satisﬁes the following conditions: (1) α is surjective in codimension one, C = α∗ B is the strict transform, and (Y, C) is a Q-factorial LC pair. (2) KY + C is nef over S. (3) Any exceptional prime divisor for α has reason to be contracted in the sense that it has positive coeﬃcient in the diﬀerence of KX + B and KY + C. More precisely, if p : V → X and q : V → Y are common resolutions such that q = α ◦ p, then p∗ (KX + B) − q ∗ (KY + C) has positive coeﬃcient at the strict transform of an arbitrary prime divisor on X whose image on Y has higher codimension. The condition 3 is referred to as the negativity of KX + B for α. If a minimal model exists, then KX + B is pseudo-eﬀective. The converse, the existence of a minimal model, is still a conjecture. The conjecture is proved to be true if the boundary B is big in [1]. A canonical model is deﬁned to be a projective morphism h : Z → S with a surjective morphism g : Y → Z with connected geometric ﬁbers from a minimal model such that KY + C = g ∗ H for an R-Cartier divisor H on Z which is ample over S. Minimal models are not necessarily unique for a given morphism f : (X, B) → S. But any minimal models are equivalent in the following sense: if αi : (X, B) ''( (Yi , Ci ) for i = 1, 2 are minimal models and pi : V → Yi are common resolutions −1 ∗ ∗ such that p−1 1 ◦ α1 = p2 ◦ α2 , then we can prove that p1 (KY1 + C1 ) = p2 (KY2 + C2 ) as a consequence of the Hodge index theorem. The negativity condition implies that the Yi are isomorphic in codimension one. On the other hand, the canonical model is unique in the following sense: the composite rational map to the canonical model β = g ◦ α is uniquely determined by the given morphism f : (X, B) → S.

3. Numerical linear systems We propose deﬁnitions of numerical linear systems and numerical Zariski decompositions. They are easier to deal with than R-linear systems as in [1] and sectional decompositions as in [4] and [7]. Let f : X → S be a projective morphism from a Q-factorial normal variety, and D an R-Cartier divisor which is pseudo-eﬀective over S, i.e., [D] ∈ Psef(X/S). We deﬁne the numerically ﬁxed part of D by N (D) = lim(inf{DD | DD ≡ D + rA, DD ≥ 0}) l↓0

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where A is an arbitrarily ﬁxed ample divisor, DD ≥ 0 means that the R-divisor DD is eﬀective, and the inﬁmum of R-divisors is deﬁned by the inﬁmums of coeﬃcients. We note that N (D) does not depend on the choice of A. We also note that there are only ﬁnitely many irreducible components of N (D), since they are linearly independent in N 1 (X/S). For example, if D is nef, then N (D) = 0. A similar statement to the following lemma for the case of R-linear systems is proved in [1] using a result in [7]. We believe that our approach is more natural and easier. Lemma 2. Let f : (X, B) → S be a projective morphism from a Q-factorial LC pair, and let α : (X, B) ''( (Y, C) be a minimal model. Then the prime divisors contracted by α are precisely the irreducible components of the numerically ﬁxed part of KX + B. ˜ → X and q : X ˜ → Y be a common resolution. Then p∗ (KX + Proof. Let p : X ∗ B) − q (KY + C) is eﬀective, and the coeﬃcients of the exceptional divisors of α are positive. Since KY + C is nef, there is no numerically ﬁxed part; inf{DYD | DYD ≡ KY + C + rAY , DYD ≥ 0} = 0 for any r > 0 and any ample divisor AY on Y . Because of the negativity of KX +B for α, the diﬀerence (KX + B + rα∗−1 AY ) − p∗ q ∗ (KY + C + rAY ) is still eﬀective if r is small enough, where α∗−1 AY is the strict transform of AY . Then it follows that inf{DD | DD ≡ KX + B + rα∗−1 AY , DD ≥ 0} is exceptional for α, and so are any irreducible components of the numerically ﬁxed part of KX + B. Conversely, if DD ≡ KX + B + rA, then α∗ DD ≡ KY + C + rα∗ A. Hence D D − p∗ q ∗ α∗ DD has positive coeﬃcients on any exceptional divisors of α if r is small enough. Therefore any exceptional divisors of α are numerically ﬁxed for KX + B.

4. Cone decompositions A polytope is a closed convex hull of ﬁnitely many points in a real vector space. It is called rational if these points have rational coordinates. We start with a polytope decomposition with respect to the canonical models: ¯ be a Q-factorial KLT pair with a projective morphism Theorem 3. Let (X, B) f : X → S to a base space, Q-Cartier divisors, and V˜ a % B1 , . . . , Br eﬀective r ∼ polytope in the space {B = i bi Bi | bi ∈ R} = R such that the pairs (X, B) are LC for all B ∈ V˜ . We consider a closed convex subset V = {B ∈ V˜ | [KX + B] ∈ Psef(X/S)}.

321 Assume that for each B ∈ V , there exist a minimal model α : (X, B) ''( (Y, C) and a canonical model g : Y → Z for f : (X, B) → S. Moreover assume that there exists a real number r > 0 for each given B ∈ V with α : X ''( Y and g : Y → Z as above, such that the morphism g : (Y, α∗ B D ) → Z for B D ∈ V˜ has minimal and canonical models whenever [KY + α∗ B D ] ∈ Psef(Y /Z) and pB D − Bp ≤ r, where pp denotes the maximum norm of the coeﬃcients. Then there exist a ﬁnite decomposition into disjoint subsets V =

s &

Vj

j=1

and rational maps βj : X ''( Zj satisfying the following conditions: (1) B ∈ Vj if and only if βj gives the canonical model for f : (X, B) → S. (2) The closures V¯j , hence V , are polytopes for all j. Moreover, if V˜ is a rational polytope, then so are the V¯j and V . Proof. We use an idea of Shokurov [8]. We proceed by induction on the dimension of V . If dim V = 0, then the assertion is clear. Assume that dim V > 0. We ﬁx an arbitrary point B0 ∈ V . Let α0 : (X, B0 ) ''( (Y0 , C0 ) be a minimal model and g0 : (Y0 , C0 ) → Z0 a canonical model. We can write KY0 + C0 = g0∗ H0 for an R-Cartier divisor H0 on Z0 which is ample over S. We take a real number r > 0 such that g0 : (Y0 , α0∗ B) → Z0 has a minimal model α : (Y0 , α0∗ B) ''( (Y, C) and a canonical model g : (Y, C) → Z with h : Z → Z0 if KY0 + α0∗ B is pseudo-eﬀective over Z0 and pB − B0 p ≤ r. If r is suﬃciently small, then KX + B is negative for α0 . We can write KY + C = g ∗ H for an R-Cartier divisor H on Z which is ample over Z0 . If we take δ > 0 suﬃciently small, then (1 − δ)h∗ H0 + δH is ample over S. If we set B D = (1 − δ)B0 + δB, then α ◦ α0 : (X, B D ) ''( (Y, C D ) for C D = (1 − δ)α∗ C0 + δC is a minimal model for f : (X, B D ) → S because the negativity still holds, and g : (Y, C D ) → Z is a canonical model because the ampleness holds. ˜ inside V˜ which contains B0 in the relative interior and We take a polytope U ˜ to be rational when which is contained in the above r neighborhood. We take U V˜ is rational. Let ˜ | [KY + α0∗ B] ∈ Psef(Y0 /Z0 )}. U = {B ∈ U 0 ˜ ⊂ U . By the induction assumption, the boundary ∂U is a union of Then V ∩ U polytopes, and there is a ﬁnite polytope decomposition of ∂U which corresponds to the classiﬁcation of canonical models of g0 : (Y0 , α0∗ B) → Z0 . Moreover it is rational if so is V˜ . It also follows that U ⊂ V . ˜ was chosen suﬃciently small, then the cones over these polytopes with If U the vertex B0 give a ﬁnite polytope decomposition of U , which is rational if V˜ is. Since V is compact, it is covered by ﬁnitely many such U ’s, and the assertion is proved.

322

Yujiro Kawamata

Next we consider a further polytope decomposition with respect to the minimal models: Theorem 4. Assume the conditions of the above theorem. Then each Vj can be decomposed further into a ﬁnite disjoint union Vj =

t &

Wj,k

k=1

which satisﬁes the following conditions: let α : X ''( Y be a birational map such that W = {B ∈ V | α is a minimal model for (X, B)} is non-empty. Then (1) There exists an index j such that W ⊂ V¯j . (2) If W ∩ Vj is non-empty for some j, then W ∩ Vj coincides with one of the Wj,k . ¯ j,k is a polytope for any j and k. Moreover, if V˜ is a rational (3) The closure W ¯ j,k . polytope, then so are the W We note that there may be inﬁnitely many W ’s such that W ∩ Vj = Wj,k for ﬁxed j, k. Proof. 1. If α∗ (KX + Bi ) is nef for i = 1, 2, then so is α∗ (KX + tB1 + (1 − t)B2 ) for t ∈ [0, 1]. The negativity descends as well, hence W is a convex subset of V . We take a point B ∈ W in the relative interior, and let g : Y → Z be the canonical model for (Y, α∗ B). Then g ∗ Nef(Z/S) is a face of Nef(Y /S), and [α∗ (KX + B)] ∈ g ∗ Amp(Z/S). Since [α∗ (KX + B D )] ∈ Nef(Y /S) for all B D ∈ W , it follows that [α∗ (KX + B D )] ∈ g ∗ Amp(Z/S) if B D is a point in the relative interior. Therefore W ⊂ V¯j if Vj corresponds to g ◦ α. 2. Let αi : X ''( Yi for i = 1, 2 be birational maps, and let Wi be the corresponding subsets of V which are assumed to be non-empty. Assume that there are morphisms gi : Yi → Z such that β = g1 ◦ α1 = g2 ◦ α2 corresponds to some Vj . Let γ : Y1 ''( Y2 be the birational map such that α2 = γ ◦ α1 . We claim that if γ is an isomorphism in codimension one, then W1 ∩ Vj = W2 ∩ Vj . Indeed, if B ∈ W1 ∩ Vj , then KY1 + α1∗ B = g1∗ H for some H on Z which is ample over S. Since γ is an isomorphism in codimension one, it follows that KY2 + α2∗ B = g2∗ H, hence KY2 + α2∗ B is nef and equivalent to KY1 + α1∗ B. The negativity for the exceptional divisors holds at the same time for Y1 and Y2 . Therefore B ∈ W2 ∩ Vj . Let {Em } be the set of all the prime divisors Em that are contained in the numerically ﬁxed part of KX + B D for B D being a vertex of V¯j . By Lemma 2, this is a ﬁnite set. Any numerically ﬁxed component of KX + B for arbitrary B ∈ Vj belongs to this set. Therefore there are only ﬁnitely many possibilities for the set

323 of prime divisors which coincides with the set of numerically ﬁxed components of KX + B for some B ∈ Vj . Hence the decomposition is ﬁnite. 3. The negativity condition of a prime divisor with respect to a birational map is expressed by a linear inequality with rational coeﬃcients. Thus each chamber Wj,k is bounded by linear equalities with rational coeﬃcients of the following two types: (a) those coming from the negativity condition for the corresponding minimal model in the case where the boundary is open, and (b) those from other minimal models in the case of closed boundaries. Hence we have the assertion. Corollary 5. A sequence of ﬂips in the MMP with scaling terminates if minimal and canonical models exist along the line segment in the space of divisors corresponding to this MMP process. Proof. The chambers Vj are ﬁnite in number. Therefore we may ﬁx one Vj in order to prove the termination. Let α : X − ''( X + be a ﬂip in the MMP for a pair (X, B) with scaling H, and let t be the real number such that KX − + B + tH is numerically trivial for the ﬂip. Suppose that both (X, B + (t + r)H) and (X, B + (t − r)H) belong to the same chamber Vj . Thus X − and X + are minimal models respectively for these pairs. Let g ± : X ± → Z be the canonical models. There are ample R-divisors H ± on Z such that KX ± + B + (t ∓ r)H = (g ± )∗ H ± . Since α is an isomorphism in codimension one, it follows that KX ∓ + B + (t ∓ r)H = (g ∓ )∗ H ± . In other words, KX ± + B + (t ± r)H is nef. But this is a contradiction. Therefore there does not exists a ﬂip inside the same chamber Vj . Corollary 6. Let f : (X, B) → S be a projective morphism from a KLT pair, and let αi : (X, B) ''( (Yi , Ci ) (i = 1, 2) be two minimal models. Assume that they have the same canonical model. Then they are connected by a sequence of ﬂops. We note that the boundary B may not be a Q-divisor. Proof. The Yi are isomorphic in codimension one by Lemma 2. Let gi : Yi → Z be the morphisms to the canonical model. Let Hi be ample R-divisors on the Yi . Let V be the triangle spanned by the R-divisors 0, H1 and H2 on Y1 , where we use the same symbol for the strict transforms as usual. We may assume that the (Y1 , Hi ) are KLT. There are ﬁnitely many subsets Vj of V corresponding to the canonical models by the theorem. If we replace the Hi by rHi for suﬃciently small r, then we may assume that all the V¯j contain 0. Then the MMP with scaling corresponding to the line segment joining KY1 + H1 and KY1 + H2 gives the desired sequence of ﬂops. We close this paper with some examples in order to illustrate the theorems. Example 7. Let f0 : X0 → S be a ruled surface with two disjoint sections S0 and S1 over a curve, α0 : X → X0 a blowing up at a point on S0 , and f : X → S the composite map. There is one singular ﬁber of f which will be denoted by C0 + C1 , where C0 is the exceptional curve for α0 .

324

Yujiro Kawamata

Let V = {B = 12 C0 + tC1 |, 0 ≤ t ≤ 1}. Then the canonical models for the (X, B) are reduced to the identity to S for all B ∈ V . Let W− , W0 and W+ be the subsets of V deﬁned by the conditions t < 12 , t = 12 and t > 21 , respectively. They correspond to minimal models α0 : X → X0 , X → X, and α1 : X → X1 , the contraction morphism of C1 . Example 8. Let X − be a Calabi-Yau threefold and let α : X − → Y ← X + be a ﬂop such that there is a non-canonical isomorphism h : X − ∼ = X + as in Example 10. − − + − Let H be an ample divisor on X , and let H = α∗ H be the strict transform which is also considered to be a divisor on X − by using h. If r > 0 is suﬃciently − + small, then (X − , Bt ) for B;t = r((1 ; −+t)H + tH ) with t ∈−[0, 1] are KLT.0 We have − 0 a decomposition V = V V V , where V = [0, 1], V = [0, 1/2), V = {1/2} and V + = (1/2, 1], corresponding to the canonical models given by X − , Y and X + . The chambers W − = [0, 1/2] and W + = [1/2, 1] correspond to minimal models X − and X + . When t = 1/2, the canonical model Y has two minimal models X ± . We recall two examples from [5] Example 3.8 in which there are inﬁnitely many cones in the space of numerical equivalence classes of R-divisors. In this sense we can say that our theorems deal with only “local” situations. Example 9. Let f : X → S be the versal deformation of a singular ﬁber of type I2 on an elliptic surface, i.e., a curve C which has two irreducible components isomorphic to P1 intersecting transversally at two points. We have dim X = 3 and dim N 1 (X/S) = 2. There are coordinates on N 1 (X/S) deﬁned by x = (D · C1 ) and y = (D · C2 ) where the Ci are the irreducible components of C. The pseudo-eﬀective cone is the half space deﬁned by x + y ≥ 0. The eﬀective cone is not closed because the points (1, −1) and (−1, 1) are not represented by any eﬀective R-divisors. The rays generated by (n + 1, −n) for n ∈ Z divide the pseudo-eﬀective cone into inﬁnitely many polytopes which correspond to canonical models. Example 10. Let X be a generic hypersurface of degrees (2, 2, 3) in P1 × P1 × P2 . The pseudo-eﬀective cone is the cone over the convex hull of the points A = (0, 0, 1) and Cn = (n + 1, −n, 32 n(n + 1)) for n ∈ Z, because its boundary points correspond to ﬁbrations to lower dimensional spaces, i.e., elliptic ﬁbrations and K3 ﬁbrations. The eﬀective cone is closed because the points A and Cn are represented by eﬀective divisors. But it is divided into inﬁnitely many polytopes again. The projection X → S = P1 × P1 induces a linear map N 1 (X) → N 1 (X/S) given by (x, y, z) ?→ (x, y), and the eﬀective cone in the previous example is the image under this map.

References [1] Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan. Existence of minimal models for varieties of log general type. math.AG/0610203. [2] Christopher Hacon math.AG/0507597.

and

James

McKernan.

On

the

existence

of

ﬂips.

325 [3] Yujiro Kawamata, Katsumi Matsuda and Kenji Matsuki. Introduction to the minimal model problem. in Algebraic Geometry Sendai 1985, Advanced Studies in Pure Math. 10 (1987), Kinokuniya and North-Holland, 283–360. [4] Yujiro Kawamata. Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces. Ann. of Math. 127 (1988), 93–163. [5] Yujiro Kawamata. On the cone of divisors of Calabi-Yau ﬁber spaces. alggeom/9701006, Internat. J. Math. 8 (1997), 665–687. [6] Yujiro Kawamata. Flops connect minimal models. arXiv:0704.1013. [7] Noboru Nakayama. Zariski-decomposition and abundance. MSJ Memoirs 14, Mathematical Society of Japan, Tokyo, 2004. xiv+277 pp. ISBN: 4-931469-31-0. [8] V. V. Shokurov. 3-fold log models. Algebraic geometry 4, J. Math. Sci. 81 (1996), no. 3, 2667–2699. Yujiro Kawamata: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan E-mail: [email protected]

p-elementary subgroups of the Cremona group of rank 3 Yuri Prokhorov∗

Abstract. For the subgroups of the Cremona group Cr3 (C) having the form (µp )s , where p is prime, we obtain an upper bound for s. Our bound is sharp if p ≥ 17. 2010 Mathematics Subject Classiﬁcation. 14E07

1. Introduction Let k be an algebraically closed ﬁeld. The Cremona group Crn (k) is the group of birational transformations of Pnk , or equivalently the group of k-automorphisms of the ﬁeld k(x1 , . . . , xn ). Finite subgroups of Cr2 (C) are completely classiﬁed (see [DI09] and references therein). In contrast, subgroups of Crn (k) for n ≥ 3 are not studied well (cf. [Pro09]). In the present paper we study a certain kind of abelian subgroups of Cr3 (C). Let p be a prime number. We say that a group G is p-elementary if G @ (µp )s for some positive integer s. In this case s is called the rank of G and denoted by rk G. Theorem 1.1 ([Bea07]). Let p be a prime = K char(k) and let G ⊂ Cr2 (k) be a p-elementary subgroup. Then: rk G ≤ 2 + δp,3 + 2δp,2 where δi,j is Kronecker’s delta. Moreover, for any such p this bound is attained for some G. These “maximal” groups G are classiﬁed up to conjugacy in Cr2 (k). More generally, instead of Crn (k) we also can consider the group Bir(X) of birational automorphisms of an arbitrary rationally connected variety X. Our main result is the following Theorem 1.2. Let X be a rationally connected threefold deﬁned over a ﬁeld of characteristic 0, let p be a prime, and let G ⊂ Bir(X) be a p-elementary subgroup. Then   7 if p = 2,  5 if p = 3, rk G ≤ (1.3)  4 if p = 5, 7, 11, or 13,    3 if p ≥ 17. ∗ The

author was partially supported by RFBR, grant No. 08-01-00395-a and Leading Scientiﬁc Schools (grants No. 1983.2008.1, 1987.2008.1).

328

Yuri Prokhorov

For any prime p ≥ 17 this bound is attained for some subgroup G ⊂ Cr3 (C). (However we do not assert that the bound (1.3) is sharp for p ≤ 13). Remark 1.4. (i) Note that Cr1 (k) @ PGL2 (k). Hence for any prime p K= char(k) and any p-elementary subgroup G ⊂ Cr1 (k), we have rk G ≤ 1 + δp,2 (see, e.g., [Bea07, Lemma 2.1]). (ii) Since Cr1 (k) × Cr2 (k) admits (a lot of) embeddings into Cr3 (k), the group Cr3 (k) contains a p-elementary subgroup G of rank 3 + δp,3 + 3δp,2 . This shows the last assertion of our theorem. The following consequence of Theorem 1.2 was proposed by A. Beauville. Corollary 1.5. The group Cr3 (C) is not isomorphic to Crn (C) for n K= 3 as an abstract group. Proof. Denote by ξ(n, p) the maximal rank of a p-elementary group contained in Crn (C). Then ξ(2, 17) = 2 < ξ(3, 17) = 3 and ξ(n, 17) ≥ n by Theorems 1.1 and 1.2. Our method is a generalization of the method used for the study of ﬁnite subgroups of Cr2 (k) [Bea07], [DI09]. Similarly to [Pro09] we use the equivariant three-dimensional minimal model program. This easily allows us to reduce the problem to the study of automorphism groups of some (not necessarily smooth) Fano threefolds. Acknowledgments. I would like to thank J.-P. Serre for asking me the question considered here and for useful comments. I am also grateful to A. Beauville for proposing me Corollary 1.5 and for his interest in my paper. This paper was written at the IHES (Bures-sur-Yvette) during my visit in 2009. I am grateful to IHES for the support and hospitality. Finally I would like to thank the referee for several suggestions that make the paper more readable and clear.

2. Preliminaries Clearly, we may assume that k = C. All the groups in this paper are multiplicative. In particular, we denote a cyclic group of order n by µn . 2.1. Terminal singularities. We need a few facts on the classiﬁcation of threedimensional terminal singularities (see [Mor85], [Rei87]). Let (X _ P ) be a germ of a three-dimensional terminal singularity. Then (X _ P ) is isolated, i.e., Sing(X) = {P }. The index of (X _ P ) is the minimal positive integer r such that rKX is Cartier. If r = 1, then (X _ P ) is Gorenstein. In this case (X _ P ) is analytically isomorphic to a hypersurface singularity in C4 of multiplicity 2. Moreover, any Weil Q-Cartier divisor D on (X _ P ) is Cartier. If r > 1, then there is a cyclic, ´etale outside of P cover π : (X q _ P q ) → (X _ P ) of degree r such that (X q _ P q ) is a Gorenstein terminal singularity (or a smooth point). This π is called the index-one cover of (X _ P ).

329 Theorem 2.2 ([Mor85], [Rei87]). In the above notation (X q _ P q ) is analytically µr -isomorphic to a hypersurface in C4 with µr -semi-invariant 1 coordinates x1 , . . . , x4 , and the action is given by (x1 , . . . , x4 ) ?−→ (εa1 x1 , . . . , εa4 x4 ) for some primitive r-th root of unity ε, where one of the following holds: (i) (a1 , . . . , a4 ) ≡ (1, −1, a2 , 0) mod r,

gcd(a2 , r) = 1,

(ii) r = 4 and (a1 , . . . , a4 ) ≡ (1, −1, 1, 2) mod 4. Deﬁnition 2.3. A G-variety is a variety X provided with a biregular faithful action of a ﬁnite group G. We say that a normal G-variety X is GQ-factorial if any G-invariant Weil divisor on X is Q-Cartier. A projective normal G-variety X is called GQ-Fano if it is GQ-factorial, has at worst terminal singularities, −KX is ample, and rk Pic(X)G = 1. Lemma 2.4. Let (X _ P ) be a germ of a threefold terminal singularity and let G ⊂ Aut(X _ P ) be a p-elementary subgroup. Then rk G ≤ 3 + δ2,p . Proof. Assume that rk G ≥ 4 + δ2,p . First we consider the case where (X _ P ) is Gorenstein. The group G acts faithfully on the Zariski tangent space TP,X , so G ⊂ GL(TP,X ), where dim TP,X = 3 or 4. If dim TP,X = 3, then G is contained in a maximal torus of GL3 (C), so rk G ≤ 3 and we are done. Thus we may assume that dim TP,X = 4. Take semi-invariant coordinates x1 , . . . , x4 in TP,X . There is a G-equivariant analytic embedding (X _ P ) ⊂ C4x1 ,...,x4 . As above, rk G ≤ 4. Thus we may assume that rk G ≤ 4 and p > 2. Let φ(x1 , . . . , x4 ) = 0 be an equation of X, where % φ is a G-semi-invariant function. Regard φ as a power series and write φ = d φd , where φd is the sum of all monomials of degree d. Since the action of G on x1 , . . . , x4 is linear, all the φd ’s are semi-invariants of the same G-weight w = wt φd . Hence, for any φd , φd" = K 0 we have d − dD ≡ 0 mod p. Since (X _ P ) is a terminal singularity, φ2 K= 0 and so φ3 = 0. Recall that G @ (µp )4 , p ≥ 3. In this case, φ2 must be a monomial. Thus up to permutations of coordinates and scalar multiplication we get either φ2 = x21 or φ2 = x1 x2 . In particular, we have rk φ2 ≤ 2 and φ3 = 0. This contradicts the classiﬁcation of terminal singularities [Mor85], [Rei87]. Now assume that (X _ P ) is non-Gorenstein of index r > 1. Consider the index-one cover π : (X q _ P q ) → (X _ P ) (see 2.1). Here (X q _ P q ) is a Gorenstein terminal point and the map X q \{P q } → X \{P } can be regarded as the topological universal cover. Hence there exists a natural lifting Gq ⊂ Aut(X q _ P q ) ﬁtting in the following exact sequence 1 −→ µr −→ Gq −→ G −→ 1.

(2.5)

It is suﬃcient to show that there exists a subgroup G• ⊂ Gq isomorphic to G (but we do not assert that the sequence splits). Indeed, in this case G• @ G 1 In

invariant theory people often say “relative invariant” rather than “semi-invariant”. We prefer to use the terminology of [Mor85].

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acts faithfully on the terminal Gorenstein singularity (X q _ P q ) and we can apply the case considered above. We may assume that Gq is not abelian (otherwise a subgroup G• @ G obviously exists). The group Gq permutes eigenspaces of µr . By Theorem 2.2 the subspace T := {x4 = 0} ⊂ C4x1 ,...,x4 is Gq -invariant and µr acts on any eigenspace T1 ⊂ T faithfully. On the other hand, by (2.5) we see that the derived subgroup [Gq , Gq ] is contained in µr . In particular, [Gq , Gq ] is abelian and also acts on any eigenspace T1 ⊂ T faithfully. Since dim T = 3, this implies that the representation of Gq on T is irreducible (otherwise T has a one-dimensional subrepresentation, say T1 , and the kernel of the map G → GL(T1 ) @ C∗ must contain [Gq , Gq ]). Hence eigenspaces of µr have the same dimension and so µr acts on T by scalar multiplication. By Theorem 2.2 this is possible only if r = 2. Let Gqp ⊂ Gq be a Sylow p-subgroup. If µr ∩ Gqp = {1}, then Gqp @ G and we are done. Thus we assume that µr ⊂ Gqp , so p = r = 2 and Gqp = Gq . But then Gq is a 2-group, so the dimension of its irreducible representation must be a power of 2. Hence T is reducible, a contradiction. Lemma 2.6. Let X be a G-threefold with isolated singularities. (i) If there is a curve C ⊂ X of G-ﬁxed points, then rk G ≤ 2. (ii) If there is a surface S ⊂ X of G-ﬁxed points, then rk G ≤ 1. If moreover S is singular along a curve, then G = {1}. Sketch of the Proof. Consider the action of G on the tangent space to X at a general point of C (resp. S). G-equivariant minimal model program. Let X be a rationally connected three-dimensional algebraic variety and let G ⊂ Bir(X) be a ﬁnite subgroup. By shrinking X we may assume that G acts on X biregularly. The quotient Y = ˆ X/G is quasiprojective, so there exists a projective completion Yˆ ⊃ Y . Let X ˆ ˆ be the normalization of Y in the function ﬁeld C(X). Then X is a projective variety birational to X admitting a biregular action of G. There is an equivariant ˜ → X, ˆ see [AW97]. Run the G-equivariant minimal resolution of singularities X ˜ → X, ¯ see [Mor88, 0.3.14]. Running this program we stay in model program: X the category of projective normal varieties with at worst terminal GQ-factorial singularities. Since X is rationally connected, on the ﬁnal step we get a Fano-Mori ¯ → Z. Here dim Z < dim X, Z is normal, f has connected ﬁbers, ﬁbration f : X the anticanonical Weil divisor −KX¯ is ample over Z, and the relative G-invariant ¯ G is one. Obviously, we have the following possibilities: Picard number ρ(X) (i) Z is a rational surface and a general ﬁber F = f −1 (y) is a conic; (ii) Z @ P1 and a general ﬁber F = f −1 (y) is a smooth del Pezzo surface; ¯ is a GQ-Fano threefold. (iii) Z is a point and X Proposition 2.7. In the above notation assume that Z is not a point. Then rk G ≤ 3 + δp,3 + 3δp,2 . In particular, (1.3) holds.

331 Proof. Let G0 ⊂ G be the kernel of the homomorphism G → Aut(Z). The group G1 := G/G0 acts eﬀectively on Z and G0 acts eﬀectively on a general ﬁber F ⊂ X of f . Hence, G1 ⊂ Aut(Z) and G0 ⊂ Aut(F ). Clearly, G0 and G1 are p-elementary groups with rk G0 + rk G1 = rk G. Assume that Z @ P1 . Then rk G1 ≤ 1 + δp,2 . By Theorem 1.1 we obtain rk G0 ≤ 2 + δp,3 + 2δp,2 . This proves our assertion in the case Z @ P1 . The case dim Z = 2 is treated similarly. 2.8. Main assumption. Thus from now on we assume that we are in the case ¯ we may assume that our original X is a GQ-Fano (iii). Replacing X with X threefold. The group G acts naturally on H 0 (X, −KX ). If H 0 (X, −KX ) K= 0, then there exists a G-semi-invariant section s ∈ H 0 (X, −KX ) (because G is an abelian group). This section gives us an invariant member S ∈ |−KX |. Lemma 2.9. Let X be a GQ-Fano threefold, where G is a p-elementary group with rk G ≥ δp,2 + 4. Let S be an invariant Weil divisor such that −(KX + S) is nef. Then the pair (X, S) is log canonical (LC ). Proof. Assume that the pair (X, S) is not LC. Since S is G-invariant and ρ(X)G = 1, we see that S is numerically proportional to KX . Since −(KX + S) is nef, S is ample. We apply quite standard connectedness arguments of Shokurov [Sho93] (see, e.g., [MP09, Prop. 2.6]): for a suitable G-invariant boundary D, the pair (X, D) is LC, the divisor −(KX + D) is ample, and the minimal locus V of log canonical singularities is also G-invariant. Moreover, V is either a point or a smooth rational curve. By Lemma 2.4 we may assume that G has no ﬁxed points. Hence, V @ P1 and we have a map ς : G → Aut(P1 ). If p > 2, then ς(G) is a cyclic group, so G has a ﬁxed point, a contradiction. Let p = 2 and let G0 = ker ς. By Lemma 2.6 rk G0 ≤ 2. Therefore rk ς(G0 ) ≥ 3. Again we get a contradiction. Lemma 2.10. Let X be a GQ-Fano threefold, where G is a p-elementary group with   if p = 2, 7 rk G ≥ 5 (2.11) if p = 3,   4 if p ≥ 5. Let S ∈ |−KX | be a G-invariant member. Then we have (i) Any component Si ⊂ S is either rational or birationally ruled over an elliptic curve. (ii) The group G acts transitively on the components of S. (iii) For the stabilizer GSi we have rk GSi ≤ δp,2 + 4. (iv) The surface S is reducible (and reduced ). Proof. By Lemma 2.9 the pair (X, S) is LC. Assume that S is normal (and irreducible). By the adjunction formula KS ∼ 0. We claim that S has at worst Du Val singularities. Indeed, otherwise by the Connectedness Principle [Sho93, Th.

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6.9] S has at most two non-Du Val points. If p > 2, these points must be G-ﬁxed. This contradicts Lemma 2.4. Otherwise p = 2 and these points are ﬁxed for an index two subgroup G• ⊂ G. Again we get a contradiction by Lemma 2.4. Thus we may assume that S has at worst Du Val singularities. Let Γ be the image of G in Aut(S). By Lemma 2.6 rk G ≤ rk Γ + 1. Let S˜ → S be the minimal resolution. ˜ induces Here S˜ is a smooth K3 surface. The natural representation of Γ on H 2,0 (S) the exact sequence (see [Nik80]) 1 −→ Γ0 −→ Γ −→ Γ1 −→ 1, where Γ0 (resp. Γ1 ) is the kernel (resp. image) of the representation of Γ on ˜ The group Γ1 is cyclic. Hence either Γ1 = {1} or Γ1 @ µp . In the second H 2,0 (S). case by [Nik80, Cor. 3.2] p ≤ 19. Further, according to [Nik80, Th. 4.5] we have  4 if p = 2    2 if p = 3 rk Γ0 ≤  1 if p = 5 or 7    0 if p > 7. Combining this we obtain a contradiction with (2.11). Now assume that S is not normal. Let Si ⊂ S be an irreducible component (the case Si = S is not excluded). Let ν : S D → Si be the normalization. Write 0 ∼ ν ∗ (KX + Si ) = KS " + DD , where DD is the diﬀerent, see [Sho93, §3]. Here DD is an eﬀective reduced divisor and the pair is LC [Sho93, 3.2]. Since S is not normal, ˜ be the crepant DD = K 0. Consider the minimal resolution µ : S˜ → S D and let D D D ˜ pull-back of D , that is, µ∗ D = D and ˜ = µ∗ (KS " + DD ) ∼ 0. K˜ + D S

˜ is again an eﬀective reduced divisor. Hence S˜ is a ruled surface. If it is not Here D rational, consider the Albanese map α : S˜ → C. Clearly α is Γ-equivariant and ˜1 ⊂ D ˜ be an α-horizontal component. By the action of Γ on C is not trivial. Let D ˜ adjunction D1 is an elliptic curve. So is C. This proves (i). If the action on components Si ⊂ S is not transitive, we have an invariant divisor S D < S. Since X is GQ-factorial and ρ(X)G = 1, we can take S D so that −(KX + 2S D ) is nef. This contradicts Lemma 2.9. So, (ii) is proved. Now we prove (iii). Let Γ be the image of GSi in Aut(Si ). By Lemma 2.6 rk GSi ≤ rk Γ + 1. If Si is rational, then we get the assertion by Theorem 1.1. Assume that Si is a birationally ruled surface over an elliptic curve. As above, let S˜i → Si be the composition of the normalization and the minimal resolution, and let α : S˜i → C be the Albanese map. Then Γ acts faithfully on S˜i and α is Γ-equivariant. Thus we have a homomorphism α∗ : Γ → Aut(C). Here rk Γ ≤ rk α∗ (Γ) + 1 + δp,2 . Note that α∗ (Γ) is a p-elementary subgroup of the automorphism group of an elliptic curve. Hence, rk α∗ (Γ) ≤ 2. This implies (iii). It remains to prove (iv). Assume that S is irreducible. By (iii) the surface S is not rational. So, S is birational to a ruled surface over an elliptic curve. By Lemma 2.6 the group G acts faithfully on S. Hence, in the above notation, rk G = rk Γ ≤ rk α∗ (Γ) + 1 + δp,2 ≤ 3 + δp,2 , a contradiction.

333

3. Proof of Theorem 1.2 3.1. In this section we prove Theorem 1.2. As in 2.8 we assume that X is a GQ-Fano threefold, where G is a p-elementary subgroup of Aut(X). First we consider the case where X is non-Gorenstein, i.e., it has at least one point of index > 1. Proposition 3.2. Let G be a p-elementary group and let X be a non-Gorenstein GQ-Fano threefold. Then  7 if p = 2,    5 if p = 3, rk G ≤ 4 if p = 5, 7, 11, 13,    3 if p ≥ 17. Proof. Let P1 be a point of index r > 1 and let P1 , . . . , Pl be its G-orbit. Here l = pt for some t with t ≥ s − δ2,p − 3, where s = rk G (see Lemma 2.4). By the orbifold Riemann-Roch formula [Rei87] and a form of the Bogomolov-Miyaoka inequality [Kaw92], [KMMT00] we have O M$ 1 rP i − < 24. rP i Since ri − 1/ri ≥ 3/2, we have 3l/2 < 24 and so ps−δ2,p −3 ≤ l < 16. This gives us the desired inequality. From now on we assume that our GQ-Fano threefold X is Gorenstein, i.e., KX is a Cartier divisor. Recall (see, e.g., [IP99]) that the Picard group of a Fano variety X with at worst (log) terminal singularities is a torsion free ﬁnitely generated abelian group (@ H 2 (X, Z)). Then we can deﬁne the Fano index of X as the maximal positive integer that divides −KX in Pic(X). Proposition-Deﬁnition 3.3 (see, e.g., [IP99]). Let X be a Fano threefold with 3 at worst terminal Gorenstein singularities. The positive integer −KX is called the 3 degree of X. We can write −KX = 2g − 2, where g is an integer ≥ 2 called the genus of X. Then dim |−KX | = g + 1 ≥ 3. Corollary-Notation 3.4. In notation 3.1 the linear system |−KX | is not empty, %N so there exists a G-invariant member S ∈ |−KX |. Write S = i=1 Si , where Si are irreducible components. Theorem 3.5 ([Nam97]). Let X be a Fano threefold with terminal Gorenstein singularities. Then X is smoothable, that is, there is a ﬂat family Xt such that X0 @ X and a general member Xt is a smooth Fano threefold of the same degree,

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Fano index and Picard number. Furthermore, the number of singular points is bounded as follows: | Sing(X)| ≤ 20 − ρ(Xt ) + h1,2 (Xt ),

(3.6)

where h1,2 (Xt ) is the Hodge number. Combining the above theorem with the classiﬁcation of smooth Fano threefolds [Isk80], [MM82] (see also [IP99]) we get the following Theorem 3.7. Let X be a Fano threefold with at worst terminal Gorenstein singularities and let Xt be its smoothing. Let g and q be the genus and Fano index of X, respectively. (i) q ≤ 4. (ii) If q = 4, then X @ P3 . (iii) If q = 3, then X is a quadric in P4 (with dim Sing(X) ≤ 0). 3 (iv) If q = 2, then ρ(X) ≤ 3 and −KX = 8d, where 1 ≤ d ≤ 7. Moreover ρ(X) = 1 if and only if d ≤ 5.

(v) If q = 1 and ρ(X) = 1, then there are the following possibilities: g

2

3

4

5

6

7

8

9

10

12

h1,2 (Xt )

52

30

20

14

10

7

5

3

2

0

Lemma 3.8. Let G be a p-elementary group and let X be a Gorenstein GQ-Fano threefold. If the linear system |−KX | is not base point free, then rk G ≤ 3 + δp,2 . Proof. Assume that Bs |−KX | K= ∅. Clearly, Bs |−KX | is G-invariant. By [Isk80], [Shi89] Bs |−KX | is either a single point or a smooth rational curve. In the ﬁrst case the assertion immediately follows by Lemma 2.4. In the second case G acts on the curve C = Bs |−KX |. Since C @ P1 , the assertion follows by Lemma 2.6. Proposition 3.9. Let G be a p-elementary group, where p ≥ 5, and let X be a Gorenstein GQ-Fano threefold. Then 6 4 if p = 5, 7, 11, 13, rk G ≤ 3 if p ≥ 17. Proof. Assume that the above inequality does not hold. We use% the notation of Si ∈7 | − KX |. 3.4. In particular, N denotes the number of components of S = 3 ByG Lemma 2.10 N = pl , where l ≥ 1. Hence p divides −KX = 2g − 2 = (−KX )2 · Si N . First we claim that ρ(X) = 1. Indeed, if ρ(X) > 1, then the natural representation of G on PicQ (X) := Pic(X) ⊗ Q is decomposed as PicQ (X) =

335 V1 ⊕ V , where V1 is a trivial subrepresentation generated by the class of −KX and V is a subrepresentation such that V G = 0. Since G is a p-elementary group, dim V ≥ p − 1. Hence, ρ(X) ≥ p ≥ 5 and by the classiﬁcation [MM82] we have two possibilities: 3 • −KX = 6(11 − ρ(X)), 5 ≤ ρ(X) ≤ 10, or 3 = 28, ρ(X) = 5. • −KX 3 K≡ 0 mod p, a contradiction. In the ﬁrst case p In the last case p = 5, so −KX 3 divides −KX only if p = 5. Then ρ(X) = 6. So, dim V = 5 and V G K= 0. Again we get a contradiction. Therefore, ρ(X) = 1. Let q be the Fano index of X. We claim that X is singular. Indeed, otherwise all the Si are Cartier divisors. Then −KX = N S1 , where N ≥ p, and so q ≥ 5. This contradicts (i) of Theorem 3.7. Hence X is singular. By Lemma 2.4 and our assumption we have | Sing(X)| ≥ p. In particular, q ≤ 2 (see Theorem 3.7). If q = 1, then by Theorem 3.7 either 2 ≤ g ≤ 10 or g = 12. Thus N = p and we get the following possibilities: (p, g) = (5, 6), (7, 8), or (11, 12). Moreover, (−KX )2 · Si = (2g − 2)/N = 2. Therefore, the restriction |−KX ||Si of the (base point free) anticanonical linear system deﬁnes either an isomorphism to a quadric Si → Q ⊂ P3 or a double cover Si → P2 . In both cases the image is rational, so we get a map Gi → Cr2 (C) whose kernel is of rank ≤ 1 by Lemma 2.6 and because p > 2. Then by Theorem 1.1 rk GSi ≤ 3. Hence, rk G ≤ 4 which contradicts our assumption. Finally, consider the case q = 2. Then −KX = 2H for some ample Cartier divisor H and d := H 3 ≤ 7. Therefore, N Si · H 2 = S · H 2 = 2d. Since ρ(X) = 1, by Theorem 3.7 we get p = d = 5. Then we apply (3.6). In this case, h1,2 (Xt ) = 0 (see [IP99]). So, | Sing(X)| ≤ 19. On the other hand, | Sing(X)| ≥ 25 by Lemma 2.4 and our assumption. The contradiction proves the proposition.

We need the following result which is a very weak form of Shokurov’s much more general toric conjecture [McK01], [Pro03]. Lemma 3.10. Let V be a smooth Fano threefold and let D ∈ |−KV | be a divisor such that the pair (V, D) is LC. Then D has at most 3 + ρ(V ) irreducible components. %n Proof. Write D = i=1 Di . If ρ(V ) = 1, then all the Di are linearly proportional: % Di ∼ ni H, where H is an%ample generator of Pic(V ). Then −KV ∼ ni H and by Theorem 3.7 we have ni = q ≤ 4. If V is a blowup of a curve on another smooth Fano threefold W , then we can proceed by induction replacing V with W . Thus we assume that V cannot be obtained by blowing up of a curve on another smooth Fano threefold. In this situation V is called primitive ([MM83]). According to [MM83, Th. 1.6] we have ρ(V ) ≤ 3 and V has a conic bundle structure f : V → Z, where Z @ P2 (resp. Z @ P1 × P1 ) % if ρ(V ) = 2 (resp. ρ(V ) = 3). Let ] be a general ﬁber. Then 2 = −KV · ] = Di · ]. Hence D has at most two f -horizontal components and at least n−2 vertical ones. Now let h : V → W be an extremal contraction other than

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f and let ]D be any curve in a non-trivial ﬁber of h. For any f -vertical component Di ⊂ D we have Di = f −1 (Γi ), where Γi ⊂ Z is a curve, so Di · ]D = Γi · f∗ ]D ≥ 0. If ρ(V ) = 2, then Di ·lD ≥ 1. Hence, −KV ·]D ≥ n−2. On the other hand, −KV ·]D ≤ 3 (see [MM83, §3]). This immediately gives us n ≤ 5 as claimed. Finally consider the case ρ(V ) = 3. Assume that n ≥ 7. Then we can take h so that ]D meets at least three f -vertical components, say D1 , D2 , D3 . As above, −KV · ]D ≥ 3 and by the classiﬁcation of extremal rays (see [MM83, §3]) h is a del Pezzo ﬁbration. This contradicts our assumption ρ(V ) = 3. Proposition 3.11. Let G be a 2-elementary group and let X be a Gorenstein GQ-Fano threefold. Then rk G ≤ 7. Proof. Assume that rk G ≥ 8. By Lemma 2.10 we have rk GSi ≤ 5. Hence, N ≥ 8. If X is smooth, then by Lemma 3.10 we have ρ(X) ≥ 5. If furthermore X @ Y ×P1 , where Y is a del Pezzo surface, then the projection X → Y must be G-equivariant. 3 This contradicts ρ(X)G = 1. Therefore, ρ(X) = 5 and −KX = 28 or 36 (see 3 [MM82]). On the other hand, −KX is divisible by N , a contradiction. Thus X is singular. Assume that | Sing(X)| ≥ 32. Then for a smoothing Xt 3 3 of X by (3.6) we have h1,2 (Xt ) ≥ 13. Since N divides −KX = −KX , using the t classiﬁcation of Fano threefolds [Isk80], [MM82] (see also [IP99]) we get: ρ(X) = 1,

3 −KX = 8,

N = 8,

| Sing(X)| = 32.

Consider the representation of G on H 0 (X, −KX ). Since 7 = dim H 0 (X, −KX ) < rk G, this representation is not faithful (otherwise G is contained in a maximal torus of GL(H 0 (X, −KX )) = GL7 (C)). Therefore, the linear system |−KX | is not very ample. On the other hand, |−KX | is base point free (see Lemma 3.8). Hence |−KX | deﬁnes a double cover X → Y ⊂ P6 [Isk80]. Here Y is a variety of degree 4 in P6 , a variety of minimal degree. If Y is smooth, then according to the Enriques theorem (see, e.g., [Isk80, Th. 3.11]) Y is a rational scroll PP1 (E ), where E is a rank 3 vector bundle on P1 . Then X has a G-equivariant projection to a curve. This contradicts ρ(X)G = 1. Hence Y is singular. In this case, Y is a cone (again by the Enriques theorem [Isk80, Th. 3.11]). If its vertex O ∈ Y is zero-dimensional, then dim TO,Y = 6. On the other hand, X has only hypersurface singularities (see 2.1). Therefore the double cover X → Y is not ´etale over O and so G has a ﬁxed point on X. This contradicts Lemma 2.4. Thus Y is a cone over a rational normal curve of degree 4 with vertex along a line. Then X cannot have isolated singularities, a contradiction. Therefore, | Sing(X)| < 32. Then for any point P ∈ Sing(X) by Lemma 2.4 we have rk GP ≥ 4. Hence the orbit of P contains 16 elements and coincides with Sing(X), i.e., the action of G on Sing(X) is transitive. Since S ∩ Sing(X) K= ∅, we have Sing(X) ⊂ S. On the other hand, our choice of S in 2.8 is not unique: there is a basis s(1) , . . . , s(g+2) ∈ H 0 (X, −KX ) consisting of eigensections. This basis gives us G-invariant divisors S (1) , . . . , S (g+2) generating | − KX |. By the above,

337 Sing(X) ⊂ S (i) for all i. Thus Sing(X) ⊂ ∩S (i) = Bs | − KX |. This contradicts Lemma 3.8. Proposition 3.11 is proved. Proposition 3.12. Let G be an 3-elementary group and let X be a Gorenstein GQ-Fano threefold. Then rk G ≤ 5. Proof. Assume that rk G ≥ 6. By Lemma 2.10 we have rk GSi ≤ 5. Hence, N ≥ 9. If X is smooth, then by Lemma 3.10 we have ρ(X) ≥ 6 and so X @ Y ×P1 , where Y is a del Pezzo surface [MM82]. Then the projection X → Y must be G-equivariant. This contradicts ρ(X)G = 1. Therefore, X is singular. By Lemma 2.4 | Sing(X)| ≥ 36−3 = 27. Hence, for a smoothing Xt of X by (3.6) we have h1,2 (Xt ) ≥ 7 + ρ(X). 3 3 . Then we use the classiﬁcation of smooth = −KX Recall that N divides −KX t Fano threefolds [Isk80], [MM82] and get a contradiction. Now Theorem 1.2 is a consequence of Propositions 3.2, 3.9, 3.11, and 3.12.

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Yuri Prokhorov: Department of Algebra, Faculty of Mathematics, Moscow State University, Moscow 117234, Russia E-mail: [email protected]