Compactifying Moduli Spaces for Abelian Varieties

286 Pages·2008·2.3 MB·English

by  Martin C. Olsson (auth.)

Lecture Notes in Mathematics 1958
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Martin C. Olsson
Compactifying Moduli
Spaces for Abelian Varieties
ABC
MartinC.Olsson
DepartmentofMathematics
UniversityofCalifornia
Berkeley,CA94720-3840
USA
[e-mail]
ISBN:978-3-540-70518-5 e-ISBN:978-3-540-70519-2
DOI:10.1007/978-3-540-70519-2
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2008931163
MathematicsSubjectClassification(2000):14K10,14K25
(cid:1)c 2008Springer-VerlagBerlinHeidelberg
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Contents
Summary......................................................VII
0 Introduction............................................... 1
1 A Brief Primer on Algebraic Stacks ....................... 7
1.1 S-groupoids ............................................ 7
1.2 Stacks ................................................. 17
1.3 Comparison of Topologies ................................ 25
1.4 Coarse Moduli Spaces.................................... 26
1.5 Rigidification of Stacks................................... 26
2 Preliminaries .............................................. 31
2.1 Abelian Schemes and Torsors ............................. 31
2.2 Biextensions ............................................ 34
2.3 Logarithmic Geometry ................................... 43
2.4 Summary of Alexeev’s Results ............................ 51
3 Moduli of Broken Toric Varieties .......................... 57
3.1 The Basic Construction .................................. 57
3.2 Automorphisms of the Standard Family over a Field ......... 67
3.3 Deformation Theory ..................................... 70
3.4 Algebraization .......................................... 75
3.5 Approximation.......................................... 76
3.6 Automorphisms over a General Base ....................... 78
3.7 The Stack K .......................................... 80
Q
4 Moduli of Principally Polarized Abelian Varieties.......... 85
4.1 The Standard Construction............................... 85
4.2 Automorphisms over a Field .............................. 93
4.3 Deformation Theory ..................................... 98
4.4 Isomorphisms over Artinian Local Rings....................110
VI Contents
4.5 Versal Families..........................................113
4.6 Definition of the Moduli Problem..........................121
4.7 The Valuative Criterion for Properness.....................121
4.8 Algebraization ..........................................125
4.9 Completion of Proof of 4.6.2 ..............................130
5 Moduli of Abelian Varieties with Higher Degree
Polarizations ..............................................135
5.1 Rethinking A .........................................135
g,d
5.2 The Standard Construction...............................138
5.3 Another Interpretation of P(cid:1)→P ........................142
5.4 The Theta Group .......................................144
5.5 Deformation Theory .....................................158
5.6 Isomorphisms without Log Structures ......................160
5.7 Algebraization of Formal Log Structures....................163
5.8 Description of the Group Hgp .............................166
S
5.9 Specialization ...........................................178
5.10 Isomorphisms in T ....................................201
g,d
5.11 Rigidification ...........................................202
5.12 Example: Higher Dimensional Tate Curve ..................207
5.13 The Case g =1 .........................................225
6 Level Structure............................................231
6.1 First Approach Using Kummer ´etale Topology ..............231
6.2 Second Approach using the Theta Group ...................237
6.3 Resolving Singularities of Theta Functions..................241
References.....................................................273
Index of Terminology..........................................277
Index of Notation .............................................279
Summary
The problem of compactifying the moduli space A of principally polarized
g
abelian varieties has a long and rich history. The majority of recent work has
focusedonthetoroidalcompactificationsconstructedoverCbyMumfordand
his coworkers, and over Z by Chai and Faltings. The main drawback of these
compactificationsisthattheyarenotcanonicalanddonotrepresentanyrea-
sonable moduli problem on the category of schemes. The starting point for
thisworkistherealizationofAlexeevandNakamurathatthereisacanonical
compactificationofthemodulispaceofprincipallypolarizedabelianvarieties.
Indeed Alexeev describes a moduli problem representable by a proper alge-
braic stack over Z which contains A as a dense open subset of one of its
g
irreducible components.
In this text we explain how, using logarithmic structures in the sense of
Fontaine, Illusie, and Kato, one can define a moduli problem “carving out”
the main component 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.
If d and g are positive integers we construct a proper algebraic stack with
finite diagonal A over Z containing the moduli stack A of abelian va-
g,d g,d
rieties with a polarization of degree d as a dense open substack. The main
features of the stack A are that (i) over Z[1/d] it is log smooth (i.e. has
g,d
toroidal singularities), and (ii) there is a canonical extension of the kernel of
the universal polarization over A to A . The stack A is obtained by a
g,d g,d g,d
certain“rigidification”procedurefromasolutiontoamoduliproblem.Inthe
cased=1thestackA isequaltothenormalizationofthemaincomponent
g,1
in Alexeev’s compactification. In the higher degree case, our study should be
viewed as a higher dimensional version of the theory of generalized elliptic
curves introduced by Deligne and Rapoport.
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 E is
t¯
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 [15]. 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 good
1,1
notion of the N–torsion subgroup of E. More precisely, if there exists a dense
open subset U ⊂ T such that the restriction f : E → U is smooth, then
U U
the finite flat U–group scheme E [N] does not extend to a finite flat group
U
scheme over T. Deligne and Rapoport solve this problem by introducing “N-
gons”whichenablethemtodefineareasonablenotionofN–torsiongroupfor
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.
M.C.Olsson,Compactifying Moduli Spaces for Abelian Varieties.LectureNotes 1
inMathematics1958.
(cid:1)c Springer-VerlagBerlinHeidelberg2008
2 0 Introduction
Let A denote the moduli space of principally polarized abelian varieties
g
ofdimensiong forsomeintegerg ≥1.ThefirstcompactificationofA overC
g
is the so-called Satake or minimal compactification A ⊂A∗ constructed by
g g
Satakein[47].ThespaceA∗isnormalbutingeneralsingularattheboundary.
g
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
g
one over Z.
Over the complex numbers such resolutions of A∗ were constructed by
g
Ash,Mumford,Rapoport,andTaiin[9]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 [22] have given a modular interpre-
tation of these compactifications using their theory of log abelian varieties).
Later Chai and Faltings [13] 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 [3] 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 [39] and work
with Nakamura [4]). 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 ([24]) 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) ThereisanaturalopenimmersionA (cid:2)→K identifyingA withadense
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 defines a fine saturated log structure MKg on Kg such that the
log stack (Kg,MKg) is log smooth over Spec(Z) with the trivial log
structure).
0 Introduction 3
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
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), a(cid:5)→[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 (cid:2)→ 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)
[36, 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 2.1.5 there is a canonical isomorphism At (cid:7)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)(cid:5)→(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 (the reader not famililar with
(A,P,L)
thenotionofrigidification maywishtoconsulttheexamples1.5.7and1.5.8).
Thus in many ways the stack T is a more basic object than A .
g,d g,d
Withthisinmind,ourapproachtocompactifyingA istofirstconstruct
g,d
anopenimmersionT (cid:2)→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