Compactifying Moduli Spaces

141 Pages·2016·1.493 MB·English

by  Paul Hacking, Radu Laza, Dragos Oprea (auth.), Gilberto Bini, Martí Lahoz, Emanuele Macrí, Paolo Stellari (eds.)

Advanced Courses in Mathematics CRM Barcelona
Paul Hacking
Radu Laza
Dragos Oprea
Compactifying
Moduli Spaces

Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Enric Ventura

More information about this series at http://www.springer.com/series/5038
Paul Hacking • Radu Laza • Dragos Oprea
Compactifying Moduli Spaces
Editors for this volume:
Gilberto Bini, Università degli Studi di Milano
Martí Lahoz, Université Paris Diderot – Paris 7
Emanuele Macrì, Northeastern University
Paolo Stellari, Università degli Studi di Milano
Paul Hacking Radu Laza
Department of Mathematics School of Mathematics
University of Massachusetts Institute for Advanced Study
Amherst, MA, USA Princeton, NJ, USA
Dragos Oprea
Department of Mathematics
University of California
La Jolla, CA, USA

ISSN 2297-0304 ISSN 2297-0312 (e lectronic )
Advanced Courses in Mathematics - CRM Barcelona
ISBN 978-3-0348-0920-7 ISBN 978-3-0348-0921-4 (eBook)
DOI 10.1007/978-3-0348-0921-4

Library of Congress Control Number: 2015960183

Mathematics Subject Classification (2010): 14D06

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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Perspectives on the Construction and Compactification of Moduli Spaces
Radu Laza
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The GIT approachto constructing moduli spaces . . . . . . . . . 3
1.1.1 Basic GIT and moduli . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Applications of GIT to moduli . . . . . . . . . . . . . . . 9
1.2 Moduli and periods . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Period maps . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Applications of locally symmetric varieties . . . . . . . . . 16
1.2.3 Comparison to GIT compactifications . . . . . . . . . . . 19
1.3 The KSBA approach to moduli spaces . . . . . . . . . . . . . . . 23
1.3.1 The KSBA approach . . . . . . . . . . . . . . . . . . . . . 24
1.3.2 Slc singularities are Du Bois . . . . . . . . . . . . . . . . . 28
1.3.3 Asymptotic stability, K-stability, and KSBA . . . . . . . . 29
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Compact Moduli Spaces of Surfaces and Exceptional Vector Bundles
Paul Hacking
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1 Moduli spaces of surfaces of general type . . . . . . . . . . . . . . 43
2.1.1 Surfaces of general type . . . . . . . . . . . . . . . . . . . 43
2.1.2 Simultaneous resolution of Du Val singularities . . . . . . 44
2.1.3 Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.4 Expected dimension . . . . . . . . . . . . . . . . . . . . . 46
2.1.5 Compactification . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.6 Stable surfaces . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.7 Semi-log canonical singularities . . . . . . . . . . . . . . . 47
2.1.8 Dualizing sheaf . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.9 The index of an slc singularity . . . . . . . . . . . . . . . 48
v
vi Contents
2.1.10 The index-one cover . . . . . . . . . . . . . . . . . . . . . 49
2.1.11 Q-Gorenstein families of stable surfaces . . . . . . . . . . 49
2.1.12 The relative dualizing sheaf . . . . . . . . . . . . . . . . . 51
2.1.13 Definition of the moduli space M of stable
K2,χ
surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Wahl singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.1 Degenerations with Wahl singularities define boundary
divisors of the moduli space MK2,χ . . . . . . . . . . . . . 54
2.2.2 Topology of Wahl degenerations. . . . . . . . . . . . . . . 55
2.3 Examples of degenerations of Wahl type . . . . . . . . . . . . . . 57
2.4 Exceptional vector bundles associated to Wahl degenerations . . 61
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.5.1 del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . . . 63
2.5.2 Godeaux surfaces . . . . . . . . . . . . . . . . . . . . . . . 65
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Notes on the Moduli Space of Stable Quotients
Dragos Oprea
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Morphism spaces and Quot schemes over a fixed curve . . . . . . 70
3.1.1 Stable maps . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.2 Quot schemes . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1.4 Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1.5 Intersections. . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 Stable quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.1 Definition of stable quotients and examples . . . . . . . . 85
3.2.2 Construction of the moduli space . . . . . . . . . . . . . . 91
3.2.3 Obstruction theory . . . . . . . . . . . . . . . . . . . . . . 93
3.2.4 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2.5 Virtually smooth morphisms and comparison
of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 Stable quotient invariants . . . . . . . . . . . . . . . . . . . . . . 102
3.3.1 Equivariant localization . . . . . . . . . . . . . . . . . . . 102
3.3.2 Local geometries . . . . . . . . . . . . . . . . . . . . . . . 105
3.3.3 Hypersurface geometries . . . . . . . . . . . . . . . . . . . 116
3.4 Wall-crossing and other geometries . . . . . . . . . . . . . . . . . 121
3.4.1 Variation of stability . . . . . . . . . . . . . . . . . . . . . 121
3.4.2 Quasimaps to GIT quotients . . . . . . . . . . . . . . . . 124
3.4.3 Quasimap invariants of semi-positive GIT targets . . . . . 129
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Foreword
This book comes from the lectures given during the conference “Compactifying
Moduli Spaces”, held in May 2013 at Centre de Recerca Matema`tica (CRM) in
Barcelona. In them, the speakers reported on recent research on moduli theory,
from different points of view.
In recent years, moduli spaces have been investigated for their diverse ap-
plications to Algebraic Geometry, Number Theory, String Theory, and Quantum
Field Theory, just to mention a few. In particular, the notion of compactification
of moduli spaces, in its various declinations, has played a crucial role in solving
several open problems and long-standing conjectures.
Thecompactificationproblemcanbeapproachedviavarioustechniques.Ge-
ometricInvariantTheory,HodgeTheory,andtheMMPcomeintoplayasdifferent
approachestoconstructandcompactifymodulispaces.Alltheseperspectivesshed
light on particular aspects of moduli problems.
In this volume, we collect three contributions, written by Radu Laza, Paul
Hacking and Dragos Oprea. In the first of them, various ways to construct and
compactify moduli spaces are presented. In the second, some questions on the
boundary of moduli spaces of surfaces are addressed via stable vector bundles on
the smooth surfaces which degenerate to boundary points. Finally, in the third
contribution the theory of stable quotients is explained, which yields meaningful
compactifications of moduli spaces of maps.
It is our hope that these lecture notes will illustrate the wide and rich va-
riety of ideas and theories, which have developed from the general problem of
understanding moduli spaces and their geometry.
Gilberto Bini
Mart´ı Lahoz
Emanuele Macr`ı
Paolo Stellari
vii
Chapter 1
Perspectives on the Construction and
Compactification of Moduli Spaces
Radu Laza
Introduction
A central theme in algebraic geometry is the construction of compact moduli
spaces with geometric meaning. The two early successes of the moduli theory –
theconstructionandcompactificationofthemodulispacesofcurvesM andprin-
g
cipallypolarizedabelianvarieties(ppavs)A –aremodelsthatwetrytoemulate.
g
While very few other examples are so well understood, the tools developed to
study other moduli spaces have led to new developments and unexpected direc-
tionsinalgebraicgeometry.Thepurposeofthesenotesistoreviewthreestandard
approachesto constructingandcompactifyingmoduli spaces:GIT, Hodgetheory,
and MMP, and to discuss various connections between them.
OneoftheoldestapproachtomoduliproblemsisGeometricInvariantTheory
(GIT).Theideaisnatural:thevarietiesinagivenclasscanbetypicallyembedded
into a fixedprojectivespace. Due to the existence ofHilbert schemes,one obtains
a quasi-projective variety X parametrizing embedded varieties of a certain class.
Forgetting the embedding amounts to considering the quotient X/G for a certain
reductive algebraic group G. Ideally, X/G would be the moduli space of varieties
of the given class. Unfortunately, the naive quotient X/G does not make sense;
it has to be replaced by the GIT quotient X//G of Mumford [105]. While X//G
is the correct quotient from an abstract point of view, there is a price to pay:
it is typically difficult to understand which are the semistable objects (i.e., the
objects parameterized by X//G) and then some of the semistable objects are too
degenerate from a moduli point of view. Nonetheless, X//G gives a projective
modelforamodulispacewithweakmodularmeaning.SincetheGITmodelX//G
is sometimes more accessible than other models, X//G can be viewed as a first
approximationof more desirable compactifications of the moduli space.
© Springer Basel 2016 1
P. Hacking et al., Compactifying Moduli Spaces, Advanced Courses
in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0921-4_1
2 Chapter 1. Perspectives on Moduli Spaces
A different perspective on moduli is to consider the variation of the coho-
mology of the varieties in the given moduli stack M. From this point of view, one
considersthehomogeneousspaceDthatclassifiestheHodgestructuresofacertain
type, and then the quotient D/Γ which corresponds to forgetting the marking of
the cohomology. The ideal situation would be a period map P: M → D/Γ which
is an isomorphism, or at leasta birational map. Results establishing the (generic)
injectivity of the period map are called “Torelli theorems”, and a fair number of
such results are known. Unfortunately, the image of P in D is typically very hard
to understand: Griffiths’ transversalitysays that the periods of algebraic varieties
vary in a constrained way, which gives highly non-trivial systems of differential
equations. Essentially, the only cases where we don’t have to deal with these is-
sues are the classical cases of ppavs and K3 surfaces, for which all our knowledge
on their moduli is obtained by this Hodge theoretic construction. Furthermore,
having a good period map gives numerous geometric consequences. The reason
for this is that the spaces D/Γ have a lot of structure that can be translated into
geometric properties. While it is advantageous to get a description of the moduli
space as a locally symmetric variety, in practice very few examples are known.
We will briefly mention some enlargementof the applicability of period map con-
structions to moduli beyond ppavs and K3s. Finally, we will review some work of
Looijenga which gives some comparison results for the case when both the GIT
and the Hodge theoretic approach are applicable. This is in some sense an ideal
situation as both geometric and structural results exist.
While the first two approaches are based on considering the properties of
smooth objects and then constructing a global moduli space, the third approach
takesa differenttack: one constructs a moduli spaceby gluing local patches. This
gives a moduli stack,and the main issue is to carefully choose degenerationssuch
that one obtains a proper and separated stack. By the valuative criteria, it suf-
fices to consider 1-parameter degenerations. From a topological point of view,
the ideal model is a semistable degeneration X/Δ, but then the central fiber is
far from unique. A fundamental insight comes from the minimal model program
(MMP): the canonical model of varieties of general type is unique. Consequently,
by allowing “mild” singularities, one obtains a unique limit for a 1-parameter
degeneration, leading to a proper and separated moduli stack, and (under mild
assumptions)evenaprojectivecoarsemodulispace.Thistheorywasdevelopedby
Koll´ar,Shepherd,Barron,andAlexeev(KSBA)withcontributionsfromotherau-
thors.TherelationshipbetweentheKSBAapproachandtheothertwoapproaches
is not well understood. We briefly review some partial results on this subject. In
onedirection, the connectionbetweenDu Boisandsemi-log-canonical(slc) singu-
laritiesgives a link between KSBA andHodge theory. In a different direction, the
Donaldson–Tian theory of K-stability establishes a connection between GIT and
KSBA stability. These topics are rapidly evolving and suggest that much is still
to be explored in moduli theory.
The overarching theme of these notes is that each approach sheds light on
a different aspect of the moduli problem under consideration. By taking together