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

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2 Natural operations with divisorial rings

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3 The main construction

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4 Lifting lemmas

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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