A finiteness result for Siegel modular threefolds [Elektronische Ressource] / von Cord Erdenberger

Cord Erdenberger

Sujets

• Mathematics

Informations

• Publié par: gottfried_wilhelm_leibniz_universitat_hannover
• Publié le: 01 janvier 2008
• Nombre de lectures: 41
• Langue: Deutsch

Extrait

A niteness result
for
Siegel modular threefolds
Der Fakult at fur Mathematik und Physik
der Gottfried Wilhelm Leibniz Universit at Hannover
zur Erlangung des Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
von
Dipl.-Math. Cord Erdenberger,
geboren am 01.03.1977 in Neustadt a. Rbge
2008Referent: Prof. Dr. Klaus Hulek, Hannover
Koreferent: Prof. Dr. Herbert Lange, Erlangen-Nurn berg
Tag der Promotion: 17. Dezember 2007Zusammenfassung
IndervorliegendenArbeituntersuchenwirdieKodairadimensionvonSiegelschen
Modulvarietaten, die sich als Quotient des Siegelschen oberen HalbraumesH g
nach diskreten Untergruppen der symplektischen Gruppe Sp(2g,Z) darstellen
lassen.InsbesonderegebenwireineexpliziteBeschreibungdernicht–kanonischen
Singularitat en, die im Inneren dieser Varietat en liegen. Im Fall g = 3 liefert dies
zusammen mit einer genauen Untersuchung der Geometrie ihrer Kompakti zie-
rungendasHauptresultatdieserArbeit,namlich,dasses—abgesehenvoneinem
technischen Detail — nur endlich viele Untergruppen von Sp(6,Z) gibt, fur die
diehierdurchgegebenenSiegelschenModulvarietatennichtvonallgemeinemTyp
sind. Dies verallgemeinert ein ahnlic hes Resultat von Borisov fur den Fall g =2,
also fur Untergruppen von Sp(4,Z).
Der Schlusse l zum Beweis dieses Theorems ist ein Resultat von Serre, welches
besagt, dass fur g 2 jede Untergruppe von endlichem Index eine Hauptkon-
gruenzuntergruppe (n) als Normalteiler enthalt . Dies gestattet uns, jede Sie-g
gelsche Modulvarietat als Quotient vonA (n), dem Modulraum der prinzipal– g
polarisierten abelschen Varietat en mit einer Level n–Struktur, nach einer Un-
tergruppe der endlichen Gruppe Sp(2g,Z/nZ) darzustellen. Dank der zahlrei-
chen Resultate, die ub er diese RaumeA (n) bekannt sind, konnen wir uns somitg
auf die Untersuchung der endlichen Quotientenabbildungen beschranken, welche
durch die Untergruppen von Sp(2g,Z/nZ) gegeben sind. Hierbei werden sowohl
geometrische als auch algebraische Methoden benot igt, um die notwendigen Be-
dingungen zu bestimmen, die garantieren, dass die zugehorigen Siegelschen Mo-
dulvarietat en von allgemeinem Typ sind.
Schlagworte: Siegelsche Modulvarietat, Modulraum, abelsche Variet at, Ko-
dairadimension
iiiivAbstract
In this thesis we study the Kodaira dimension of Siegel modular varieties which
are obtained by taking quotients of the Siegel upper half spaceH by discreteg
subgroups of the symplectic group Sp(2g,Z). In particular, we give an explicit
description of the non–canonical singularities in the interior of these varieties.
For g = 3 this and a careful analysis of the geometry of their compacti cations
yield the main result of this thesis, namely that there are up to a technical
issue only nitely many subgroups of Sp(6 ,Z) for which the corresponding Siegel
modular variety is not of general type. This generalizes a similar niteness result
by Borisov for g =2, i.e. for subgroups of Sp(4,Z).
The key to our proof is a result due to Serre, which shows that for g 2 ev-
ery subgroup of nite index contains a principal congruence subgroup (n) asg
a normal subgroup. This allows us to exhibit every Siegel modular variety as a
quotientofA (n),themodulispaceofprincipallypolarizedabelianvarietieswithg
a level n–structure, by a subgroup of the nite group Sp(2 g,Z/nZ). Thanks to
various results on the well–studied spacesA (n), we can then con ne ourselvesg
to the study of the nite quotient maps given by the subgroups of Sp(2 g,Z/nZ).
Here both geometric and algebraic techniques are required to determine the con-
ditions to be imposed on these subgroups to ensure that the corresponding Siegel
modular varieties are of general type.
Keywords: Siegel modular variety, moduli space, abelian variety, Kodaira di-
mension
vviPreface
Siegel modular varieties are obtained by taking quotients of the Siegel upper
half spaceH by discrete subgroups of the symplectic group Sp(2g,R). Theyg
admit an interpretation as moduli spaces for abelian varieties with certain extra
data, such as polarizations and level structures. These quasi–projective normal
varieties have an analytic realization as locally symmetric varieties which allows
them to be compacti ed in various ways, in particular by the method of toroidal
compacti cation (cf. [AMRT]). One of the rst questions to ask towards a clas-
si cation of these varieties is the one for their Kodaira dimension, which is an
important birational invariant.
The focus of this thesis lies on the Kodaira dimension of Siegel modular varieties
de ned by subgroups of Sp(2 g,Z), i.e. the ones with integer coe cients. There
are quite a number of results on the moduli spaceA of principally polarizedg
abelian varieties which is de ned by the full symplectic group Sp(2 g,Z). It is
rational for g = 2 and g = 3 (cf. [Igu1], [Kat]) and unirational for g = 4 and
g =5 (cf. [Cle], [Don], [MM]). On the other hand Tai showed in [Tai] thatA isg
of general type for g 9 which was improved later by Freitag (cf. [Fre] for g 8)
and Mumford to g 7 (cf. [Mum2]). The only open case isA , whose Kodaira6
dimension is still unknown.
Not nearly as much is known for subgroups of Sp(2g,Z). While there are some
resultsforcertainfamiliesofsubgroupssuchastheonede ningthemodulispaces
A (n) parameterizing abelian varieties with level–n structure, there are almostg
no results for arbitrary subgroups. However, for g = 2, Borisov has shown that
there are only nitely many subgroups of Sp(4 ,Z) of nite index such that the
corresponding moduli spaces are not of general type (cf. [Bor]). His proof was
inspired by the work of Thompson who proved a similar statement for arithmetic
subgroups of PSL(2,R) (cf. [Tho]).
It is conjectured that Borisov’s result can be generalized to arbitrary genusg. To
dothis,agoodunderstandingofthegeometryoftheSiegelmodularvarietiesand
theircompacti cationsisrequired. Furthermore,onealsohastostudycloselythe
subgroups of Sp(2g,Z) to be able to analyze the singularities occurring both in
the varieties themselves and in their compacti cations. Thus both geometric and
viiviii
algebraic techniques are needed to acquire a complete picture of these varieties
for arbitrary subgroups of Sp(2g,Z).
In this thesis we will mostly focus on the case g = 3, where we will apply these
techniques to obtain a generalization of Borisov’s result to subgroups of Sp(6,Z)
up to a technical issue. However, quite a few of our intermediate results, which
are of interest in their own right, are given for arbitrary g, e.g. the description of
the non–canonical singularities lying in the interior, or can easily be generalized
to higher genus.
Asalreadymentionedintheabstract, thekeyideaoftheproofofthemainresult
involves a result of Serre (cf. [BLS]), which shows that for g 2 every sub-
groupof niteindexinSp(2 g,Z)isinfactacongruencesubgroup, i.e.itcontains
a principal congruence subgroup (n) for some level n. Thus Siegel modularg
varieties can be studied by considering quotients of the moduli spaceA (n) ofg
principally polarized abelian varieties with a level n–structure by suitable sub-
groups of Sp(2g,Z/nZ). Since the spacesA (n) are known to be of general typeg
for su ciently big n (cf. Theorem 2.9 for a precise bound), it su ces to study
the action of Sp(2g,Z/nZ) on them to determine the necessary conditions to be
imposed on its subgroups such that the quotients are of general type, too.
Inthe rstchapterwewillprovidethenecessarybackgroundandtoolsforstudy-
ing Siegel modular varieties and their Kodaira dimension. In particular, we will
present the method of toroidal compacti cation which can be used to compact-
ify these quasi–projective varieties. Moreover, we will introduce modular forms
which play a key role when one wants to study pluricanonical forms on these
compacti cations to determine their Kodaira dimension.
The reader familiar with these topics might want to skip this chapter and im-
mediately jump to Chapter 2, where we will state the main result of this thesis
and give a rough outline of its proof which will be carried out in the following
chapters.
Chapter 3 provides a characterization of the boundary components of the
Vorso–called Voronoi compacti cation A (n) ofA (n) in terms of isotropic sub-gg
2gmodules of (Z/nZ) .
Thefollowingthreechapterseachcontainadi erentpartoftheproofasdescribed
in the outline in Chapter 2. While Chapter 4 studies the rami cation divisors of
VorthequotientmapsonA (n)de nedbythesubgroupsofSp(2 g,Z/nZ),chaptersg
5 and 6 mostly apply algebraic techniques to get a description of the elements
in Sp(6,Z/nZ) which cause non–canonical singularities in the modular varieties
and in the boundary of their compacti cations respectively.
In the last chapter, Chapter 7, we put all the results of the previous chapters
together to form the proof of the main result.ix
Throughout this thesis we will use the following notations and conventions:
Weworkoverthe eldofthecomplexnumbers C. Inparticular,allvarieties
are de ned over C.
Thekk unit matrix is denoted by1 . If the dimensions are clear, we alsok
sometimes write just1.
When it is not ambiguous, we will often omit the pullback map when con-
sidering the pullback of a divisor. Thus, if D is a divisor on a variety X,
f f is a map from X to X, and E is a divisor on X, we abuse notation and