Classification of Algebraic Varieties

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

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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.

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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

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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.

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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

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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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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.

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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 .

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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.

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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

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(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 ,

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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.

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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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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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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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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].

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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.

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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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Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers. Comm. Algebra 31 (2003), 3547–3618. Special issue in honor of Steven L. Kleiman.

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Dan Abramovich, Martin Olsson, and Angelo Vistoli, Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58 (2008), 1057–1091.

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Dan Abramovich, Martin Olsson, and Angelo Vistoli, Twisted stable maps to tame Artin stacks. Preprint arXiv:0801.3040v2, 2008, to appear in J. Algebraic Geom.

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Yujiro Kawamata, Deformations of canonical singularities. J. Amer. Math. Soc. 12 (1999), 85–92.

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Yujiro Kawamata, Index 1 covers of log terminal surface singularities. J. Algebraic Geom. 8 (1999), no. 3, 519–527.

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[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´ anos Koll´ ar, Hulls and husks. Preprint arXiv:0801.3040v2, 2008.

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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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Noboru Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures. Topology 25 (1986), 237–251.

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Miles Reid, Canonical 3-folds. In Journ´ees de G´eometrie Alg´ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pages 273–310. Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.

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Miles Reid, Young person’s guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 345–414. Amer. Math. Soc., Providence, RI, 1987.

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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

44

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

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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”.

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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 ).

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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),

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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

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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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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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Burniat surfaces I

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

58

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

59

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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I. Bauer, F. Catanese

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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Burniat surfaces I



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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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 .

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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.

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

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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.

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

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

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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.

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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.

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

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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.

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

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

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

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

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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).

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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

87

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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(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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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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# 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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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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#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

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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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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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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.

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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

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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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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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C. Birkar, M. P˘ aun

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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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 remar