1 Summary. The problem of compactifying the moduli space A of principally po- g larized abelian varieties has a long and rich history. The majority of recent work has focused on the toroidal compactifications constructed over C by Mumford and his coworkers, and over Z by Chai and Faltings. The main drawback of these com- pactifications is that they are not canonical and do not represent any reasonable moduli problem on the category of schemes. The starting point for this work is the realization of Alexeev and Nakamura that there is a canonical compactification of themodulispaceofprincipallypolarizedabelianvarieties.IndeedAlexeevdescribes a moduli problem representable by a proper algebraic stack over Z which contains A as a dense open subset of one of its irreducible components. g Inthistextweexplainhow,usinglogarithmicstructuresinthesenseofFontaine, Illusie, and Kato, one can define a moduli problem “carving out” the main compo- nent in Alexeev’s space. We also explain how to generalize the theory to higher degree polarizations and discuss various applications to moduli spaces for abelian varieties with level structure. Ifd,g,≥1areintegersweconstructaproperalgebraicstackwithfinitediagonal A overZcontainingthemodulistackA ofabelianvarietieswithapolarization g,d g,d of degree d as a dense open substack. The main features of the stack A are that g,d (i) over Z[1/d] it is log smooth (i.e. has toroidal singularities), and (ii) there is a canonicalextensionofthethetagroupoverA toA .ThestackA isobtained g,d g,d g,d by a certain “rigidification” procedure from a solution to a moduli problem. In the case d = 1 the stack A is equal to the normalization of the main component in g,1 Alexeev’scompactification.Inthehigherdegreecase,ourstudyshouldbeviewedas a higher dimensional version of the theory of generalized elliptic curves introduced by Deligne and Rapoport. Martin C. Olsson Canonical compactifications of moduli spaces for abelian varieties February 8, 2007 Contents 0 Introduction............................................... 5 1 Preliminaries .............................................. 11 1.1 Abelian schemes and torsors .............................. 11 1.2 Biextensions ............................................ 14 1.3 Logarithmic Geometry ................................... 22 1.4 Summary of Alexeev’s results ............................. 29 1.5 Rigidification of stacks ................................... 32 2 Moduli of broken toric varieties............................ 35 2.1 The basic construction ................................... 35 2.2 Automorphisms of the standard family over a field........... 45 2.3 Deformation theory...................................... 49 2.4 Algebraization .......................................... 53 2.5 Approximation.......................................... 55 2.6 Automorphisms over a general base........................ 56 2.7 The stack K ........................................... 58 Q 3 Moduli of principally polarized abelian varieties ........... 63 3.1 The standard construction................................ 63 3.2 Automorphisms over a field............................... 71 3.3 Deformation theory...................................... 76 3.4 Isomorphisms over Artinian local rings ..................... 88 3.5 Versal families .......................................... 91 3.6 Definition of the moduli problem .......................... 99 3.7 The valuative criterion for properness ...................... 99 3.8 Algebraization ..........................................103 3.9 Completion of proof of 3.6.2 ..............................108 4 Contents 4 Moduli of abelian varieties with higher degree polarizations.113 4.1 Rethinking A .........................................113 g,d 4.2 The standard construction................................116 4.3 Another interpretation of Pf→P.........................119 4.4 The Theta group ........................................122 4.5 Deformation Theory .....................................136 4.6 Isomorphisms without log structures .......................138 4.7 Algebraization of formal log structures .....................141 4.8 Description of the group Hgp .............................144 S 4.9 Specialization ...........................................155 4.10 Isomorphisms in T ....................................178 g,d 4.11 Rigidification ...........................................180 5 Level Structure............................................187 5.1 First approach using Kummer ´etale topology................187 5.2 Second approach using the theta group.....................190 5.3 Resolving singularities of theta functions ...................193 References.....................................................221 0 Introduction In attempting to study any moduli space M, one of the basic first steps is to find a good compactification M ⊂ M. Preferably the compactification M shouldhavereasonablegeometricproperties(i.e.smoothwithM−M adivisor withnormalcrossings),andthespaceM shouldalsohaveareasonablemoduli interpretation with boundary points corresponding to degenerate objects. Probably the most basic example of this situation is the moduli space M classifying elliptic curves and variant spaces classifying elliptic curves 1,1 with level structure. For M the compactification M is the stack which 1,1 1,1 to any scheme T associates the groupoid of pairs (f : E → T,e), where f : E → T is a proper flat morphism and e : T → Esm is a section into the smooth locus of f such that for every geometric point t¯→ T the fiber Et¯ is either a genus 1 smooth curve or a rational nodal curve. InordertogeneralizethiscompactificationM ⊂M tomodulispaces 1,1 1,1 Y classifying elliptic curves with Γ–level structure for some arithmetic sub- Γ group Γ ⊂ SL (Z), Deligne and Rapoport introduced the notion of “gener- 2 alized elliptic curves” in [11]. The main difficulty is that for a scheme T, an integer N ≥ 1, and an object (f : E → T,e) ∈ M (T) there is no notion 1,1 of the N–torsion subgroup of E. More precisely, if there exists a dense open subsetU ⊂T suchthattherestrictionf :E →U issmooth,thenthefinite U U flatU–groupschemeE [N]doesnotextendtoafiniteflatgroupschemeover U T. Deligne and Rapoport solve this problem by introducing “N-gons” which enable them to define a reasonable notion of N–torsion group for degenerate objects. The moduli space M has two natural generalizations. First one can 1,1 let the genus and number of marked points vary which leads to the moduli spaces M of genus g curves with n-marked points. These spaces of course g,n have modular compactifications M ⊂ M defined by Deligne, Mumford g,n g,n and Knudsen. The second generalization of M is moduli spaces for higher 1,1 dimensional (polarized) abelian varieties. Constructing compactifications of moduli spaces for polarized abelian varieties has historically been a much more difficult problem. 6 0 Introduction Let A denote the moduli space of principally polarized abelian varieties. g The first compactification of A over C is the so-called Satake or minimal g compactification A ⊂ A∗ constructed by Satake in [41]. The space A∗ is g g g normal but in general singular at the boundary. The basic question following the construction of the Satake compactification is then how to resolve the singularities of A∗ and to generalize the theory to one over Z. g Over the complex numbers such resolutions of A∗ were constructed by g Ash,Mumford,Rapoport,andTaiin[7]wheretheyconstructedtheso-called toroidal compactifications of A . These compactifications are smooth with g boundary a divisor with normal crossings. Unfortunately, these compactifica- tionsarenotcanonicalandthereisnosimplemodularinterpretation(though recently Kajiwara, Kato, and Nakayama [18] have given a modular interpre- tation of these compactifications using their theory of log abelian varieties). Later Chai and Faltings [10] extended the toroidal compactifications to Z. The problem remained however to define a compactification of A with g a simple modular interpretation, and to generalize the theory of Deligne– Rapoport to also give modular compactifications of moduli spaces for abelian varieties with level structure and higher degree polarizations. This is the pur- pose of this text. The starting point for our work is the paper [1] in which Alexeev studied moduli of varieties with action of semi–abelian schemes (Alexeev’s work in turn built on the work of several people including Namikawa [33] and work with Nakamura [2]). He constructed compact moduli spaces for two basic moduli problems, one of which leads to a functorial compactification of the moduli space of principally polarized abelian varieties A . One feature of his g approach, is that the resulting moduli spaces have many irreducible compo- nents with one “distinguished” component containing A . One of the main g ideas in this text is that using logarithmic geometry in the sense of Fontaine and Illusie ([20]) one can give a relatively simple functorial description of the normalizations of the main components. In fact this idea can also be applied to give a modular interpretation of Alexeev’s moduli spaces of “broken toric varieties”. In the principally polarized case our work yields an Artin stack K with g the following properties: (i) The diagonal of K is finite and K is proper of Spec(Z). g g (ii) There is a natural open immersion A ,→K identifying A with a dense g g g open substack of K . g (iii) There is a good “analytic theory” at the boundary of A in K general- g g izing the theory of the Tate curve for elliptic curves. (iv) The stack K has only toroidal singularities (in fact the complement g Kg\Ag definesafinesaturatedlogstructureMKg onKg suchthatthelog stack(Kg,MKg)islogsmoothoverSpec(Z)withthetriviallogstructure). In order to study moduli of abelian varieties with higher degree polariza- tions and level structure, we need a different point of view on how to classify 0 Introduction 7 abelian schemes with polarization. Let g and d be positive integers and let A denote the moduli stack classifying pairs (A,λ) where A is an abelian g,d schemeandλ:A→At isapolarizationofdegreed(byconventionthismeans that the kernel of λ is a finite flat group scheme of rank d2). The stack A g,d can be viewed as follows. Let T denote the stack over Z associating to any g,d scheme S the groupoid of triples (A,P,L), where A is an abelian scheme over S of relative dimension g, P is a A–torsor, and L is an ample line bundle on P such that the map λ :A→Pic0(P), a7→[t∗L⊗L−1] (0.0.0.1) L a has kernel a finite flat group scheme of rank d2, where t : P → P denotes a the action on P of a (scheme-valued) point a∈A. We will show that T is g,d in fact an algebraic stack over Z. For such a triple (A,P,L) over a scheme S, let G denote the group (A,P,L) of automorphisms of the triple (A,P,L) which are the identity on A. That is, G is the group scheme classifying pairs (β,ι), where β : P → P (A,P,L) is an automorphism commuting with the A–action and ι : β∗L → L is an isomorphismoflinebundlesonP.WecallthisgroupG thetheta group (A,P,L) of (A,P,L). There is a natural inclusion G ,→ G sending u ∈ G to m (A,P,L) m theelementwithβ =idandι=u.ThisinclusionidentifiesG withacentral m subgroup of G and we write H for the quotient. If the torsor P (A,P,L) (A,P,L) is trivial, then the group G is the theta group in the sense of Mumford (A,P,L) [30, part I, §1]. In particular, by descent theory the group scheme H is (A,P,L) a finite flat group scheme of rank d2 over S. As explained in 1.1.5 there is a canonical isomorphism At ’Pic0(P). The map 0.0.0.1 therefore induces a polarization of degree d on A. This defines a map π :T →A , (A,P,L)7→(A,λ ). g,d g,d L For any object (A,P,L) ∈ T (S) (for some scheme S), the kernel of the g,d morphism of group schemes Aut (A,P,L)→Aut (π(A,P,L)) T A g,d g,d is precisely the group scheme G . This implies that one can obtain A (A,P,L) g,d by a purely stack-theoretic construction called rigidification which “kills off” the extra automorphisms provided by G . Thus in many ways the stack (A,P,L) T is a more basic object than A . g,d g,d Withthisinmind,ourapproachtocompactifyingA istofirstconstruct g,d anopenimmersionT ,→T andanextensionofthethetagroupoverT g,d g,d g,d to an extension of a finite flat group scheme of rank d2 by G over the stack m T . The stack T should be viewed as a compactification of T , though g,d g,d g,d of course it is not compact (not even separated) since the diagonal is not proper.WecanthenapplytherigidificationconstructiontoT withrespect g,d to the extension of the theta group to get a compactification A ,→ A . g,d g,d 8 0 Introduction The stack A is proper over Z with finite diagonal, A ,→ A is a g,d g,d g,d dense open immersion, and over Z[1/d] the stack A is log smooth (so it g,d has toroidal singularities). Moreover, over A there is a tautological finite g,d flat group scheme H → A whose restriction to A is the kernel of the g,d g,d universal polarization of degree d2. Remark 0.0.1.The stack A is canonically isomorphic to K . However, g,1 g because the moduli interpretation of these two stacks are different (the defi- nition of K avoids the rigidification procedure by a trick that only works in g the principally polarized case) we make the notational distinction. The text is organized as follows. Because of the many technical details involved with our construction in full generality, we have chosen to present the theory by studying in turn three moduli problems of increasing technical difficulty(brokentoricvarieties,principallypolarizedabelianvarieties,abelian varieties with higher degree polarizations). Inchapter1wesummarizethenecessarybackgroundmaterialfortherest of the text. We state our conventions about semi-abelian schemes, review the necessary theory of biextensions, and summarize the material we need from logarithmicgeometry[20]andfortheconvenienceofthereaderwerecallsome of Alexeev’s results from [1]. Chapter 2 is devoted to moduli of “broken toric varieties”. This chapter is independent of the other chapters. We have included it here because it illustrates many of the key ideas used for the moduli of abelian varieties withoutmanyofthetechnicaldetails.Themoduliproblemsconsideredinthis chapter have also been studied extensively in other contexts (see for example [22] and [24]), so this chapter may be of independent interest. In chapter 3 we turn to the problem of compactifying the moduli space of principally polarized abelian varieties. Though we subsequently will also study higher degree polarizations and level structure, we first consider the principally polarized case which does not require the more intricate theory of the theta group and is more closely related to Alexeev’s work. In chapter 4 we then turn to the full theory. The main ingredient needed to generalize the principally polarized case is to study in detail degenerations of the theta group. Finallyinchapter5weexplainhowtoconstructcompactmodulispacesfor abelian varieties with level structure. We present two approaches. One using thetheoryoflogarithmic´etalecohomology,andthesecondusingthetheoryof the theta group developed in 4. The second approach has the advantage that itlends itselftoa study ofreductions of moduli spaces atprimes dividing the level. We intend to discuss this in future writings. We also discuss in detail how to construct modular compactifications of the moduli spaces for abelian varieties with “theta level structure” defined by Mumford in [30]. 0.0.2(Acknowledgements). The author is grateful to V. Alexeev and S. Keelforseveralhelpfulconversations.Theauthoralsowouldliketothankthe 0 Introduction 9 InstituteforAdvancedStudywherepartofthisworkwasdoneforitsexcellent workingconditions,andtheAmericanInstituteofMathematicswhichhosted an excellent workshop which helped initiate this project. The author was partially supported by an NSF post–doctoral research fellowship, NSF grant DMS-0555827, and an Alfred P. Sloan research fellowship.