Geometry of Riemann Surfaces and Their Moduli Spaces

415 Pages·2009·2.253 MB·English

by  Lizhen Ji, Scott A. Wolpert, Shing-Tung Yau

Surveys in Differential Geometry
Vol. 1: Lectures given in 1990
edited by S.-T. Yau and H. Blaine Lawson
Vol. 2: Lectures given in 1993
edited by C.C. Hsiung and S.-T. Yau
Vol. 3: Lectures given in 1996
edited by C.C. Hsiung and S.-T. Yau
Vol. 4: Integrable systems
edited by Chuu Lian Terng and Karen Uhlenbeck
Vol. 5: Differential geometry inspired by string theory
edited by S.-T. Yau
Vol. 6: Essay on Einstein manifolds
edited by Claude LeBrun and McKenzie Wang
Vol. 7: Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer
edited by S.-T. Yau
Vol. 8: Papers in honor of Calabi, Lawson, Siu, and Uhlenbeck
edited by S.-T. Yau
Vol. 9: Eigenvalues of Laplacians and other geometric operators
edited by A. Grigor’yan and S-T. Yau
Vol. 10: Essays in geometry in memory of S.-S. Chern
edited by S.-T. Yau
Vol. 11: Metric and comparison geometry
edited by Jeffrey Cheeger and Karsten Grove
Vol. 12: Geometric flows
edited by Huai-Dong Cao and S.-T. Yau
Vol. 13: Geometry, analysis, and algebraic geometry
edited by Huai-Dong Cao and S.-T.Yau
Vol. 14: Geometry of Riemann surfaces and their moduli spaces
edited by Lizhen Ji, Scott A. Wolpert, and S.-T. Yau
Volume XIV
Surveys in
Differential Geometry
Geometry of Riemann surfaces
and their moduli spaces
edited by
Lizhen Ji, Scott A. Wolpert, and Shing-Tung Yau
International Press
www.intlpress.com
Series Editor: Shing-Tung Yau

Surveys in Differential Geometry, Volume 14
Geometry of Riemann surfaces and their moduli spaces

Volume Editors:
Lizhen Ji, University of Michigan
Scott A. Wolpert, University of Maryland
Shing-Tung Yau, Harvard University

2010 Mathematics Subject Classification. 53-XX, 14H10, 14H15, 14H45, 32G15, 53C43,
53C56, 58D27.
Copyright © 2009 by International Press
Somerville, Massachusetts, U.S.A.

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SurveysinDifferentialGeometryXIV
Preface
The research of Pierre Deligne and David Mumford was reported on in
The irreducibility of the space of curves of given genus, published in 1969,
pages 75–109, tome 36 of Publications Math´ematiques de L’I.H.E´.S. In the
subsequentMathematicalReview,ManfredHerrmanwrotethattheauthors
providetwoseparateproofsoftheirreducibilityofthemodulispaceofcurves
over algebraically closed fields of arbitrary characteristic. The main tool
for the first proof is a stable reduction theorem for abelian varieties, due
to Grothendieck and Mumford, applied to curves. The authors then follow
ideas of Severi and Grothendieck for the conclusion and consider properties
of the subscheme of tricanonically embedded stable curves in the Hilbert
scheme. The second proof is based on Mumford’s earlier research and the
new notion of an algebraic stack. The authors apply concepts from the the-
ory of schemes to algebraic stacks to show that the moduli stack over SpecZ
is a proper, smooth and separated stack of finite type. From a generaliza-
tionoftheEnriques-Zariskiconnectednesstheoremtheauthorsconcludethe
irreducibilityofthemodulispaceofcurves,includingthecaseofhigherlevel
moduli spaces.
IntheirintroductionDeligneandMumfordcitearangeofapproachesto
irreducibility. The approaches include a proof by Enriques-Chisini based on
analyzing fixed degree coverings of P1 with fixed number of ordinary branch
points,ananalyticproofusingTeichmu¨ller’stheoremthatTeichmu¨llerspace
is a topological cell and a proof by Grothendieck using the given stable
reduction theorem and etale cohomology. Explanation is provided on the
use of a category larger than the category of schemes. It is explained for the
new 2-category, that for objects X,Y then hom(X,Y) is a category with
all morphisms being isomorphisms. The objects of a 2-category are called
algebraic stacks and the moduli space of curves is the underlying coarse
variety of a moduli stack.
Thepresentcollectionofpapersisinhonorofthefortiethanniversaryof
theresearchofDeligneandMumford.Theirworkcontinuestothepresentas
a fundamental contribution. Over the ensuing period, families of Riemann
surfaces and algebraic curves have continued to arise in new and different
areas of mathematics and physics. The range of approaches for studying
v
vi PREFACE
families of Riemann surfaces and algebraic curves has subsequently grown
exponentially. A survey of current research on families and the moduli space
of Riemann surfaces and algebraic curves would require a series of volumes.
As editors, we set the goal of combining a collection of highly distinguished
articles discussing a sampling of current research. A consideration for the
goal was to include articles presenting algebraic, analytic and topological
approaches. We also sought to combine articles representing the present
stateoftechniquesfromthetimeofDeligneandMumford,aswellasarticles
representing approaches developed since that time.
There have been numerous influencing developments since the original
research of Deligne and Mumford. The reader can find these developments
and the continuing refinement of the existing methods of enumerative geom-
etry, intersection theory and Brill-Noether theory throughout the contribu-
tions. The Maximal Rank Conjecture for the Hilbert function of a curve
continues to guide considerations. Harer’s calculation of the second homol-
ogy group and stability of homology for the moduli space continues as a
basic influence for topological considerations. More generally, Looijenga’s
conjecture for the moduli space of to be a union of g − 1 affine subsets
influences current perspective. The Madsen-Weiss theorem that the stable
cohomology of the mapping class groups is a polynomial algebra and the
resolution of the Mumford conjecture on stable cohomology are continuing
influences. Solutions of Witten’s conjecture, the introduction of hierarchies
and even combinatorial methods to study tautological intersection numbers
are major themes for current research. The Harris and Mumford result that
for large genus the moduli space of stable curves is of general type is a
further continuing influence. The question of birationality for small gen-
era moduli spaces continues to guide investigations. Algebraic developments
have been matched by analytical and further topological approaches. Har-
monic maps have developed as a major tool. An important development is
the discovery that a harmonic map from a compact Ka¨hler manifold to a
hyperbolic Riemann surface induces a holomorphic foliation on the domain.
Morebroadly,existenceandcomparisonargumentsfornonlinearPDEshave
undergone dramatic development. The development is suggested in the Liu-
Sun-Yau results on positivity and equivalence of the classical invariant met-
rics for Teichmu¨ller space. Harer’s results and the original proof of Witten’s
conjecture involve the action of the mapping class group on Teichmu¨ller
space. Developments have been accompanied by Thurston’s foundational
theory of measured foliations, measured geodesic laminations and the clas-
sification of elements of the mapping class group. Our understanding of the
Teichmu¨ller and Weil-Petersson metrics has undergone a renaissance in the
ensuing period. Recent influencing results on the Teichmu¨ller metric geom-
etry include McMullen’s proof in genus 2 that the closure of the GL+(2;R)
action on the Hodge bundle over moduli space is an algebraic orbifold; the
Sinai-typeasymptoticsofEskinandMirzakhaniforcountingpseudoAnosov
PREFACE vii
elementsinthemappingclassgroupandaprecisepictureoftheTeichmu¨ller
metric in finite and infinite dimensions.
Thevolumerepresentsacompilationofthetimeandeffortoftheauthors.
Wewouldliketotakethisopportunitytothankeachauthorfortheircontri-
bution. We also take this opportunity to thank the referees for their impor-
tant role. We invite the reader to study each of the articles.
Lizhen Ji
Scott A. Wolpert
Shing-Tung Yau
October, 2009
SurveysinDifferentialGeometryXIV
Contents
Preface......................................................... v
Divisors in the moduli spaces of curves
Enrico Arbarello and Maurizio Cornalba........................ 1
Stability phenomena in the topology of moduli spaces
Ralph L. Cohen................................................. 23
Birational aspects of the geometry of M
g
Gavril Farkas................................................... 57
The universal Whitham hierarchy and the geometry of
the moduli space of pointed Riemann surfaces
Samuel Grushevsky and Igor Krichever.......................... 111
Brill-Noether theory
Joe Harris...................................................... 131
GL+(R)-orbit closures via topological splittings
2
Pascal Hubert, Erwan Lanneau, and Martin Mo¨ller ............. 145
Harmonic mappings and moduli spaces of Riemann
surfaces
Ju¨rgen Jost and Shing-Tung Yau................................ 171
Algebraic structures on the topology of moduli spaces
of curves and maps
Y.-P. Lee and R. Vakil.......................................... 197
Recent development on the geometry of the Teichmu¨ller
and moduli spaces of Riemann surfaces
Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau................. 221
The universal properties of Teichmu¨ller spaces
Vladimir Markovic and Dragomir Sˇari´c ......................... 261
x CONTENTS
Geometry of Teichmu¨ller space with the Teichmu¨ller
metric
Howard Masur.................................................. 295
GIT constructions of moduli spaces of stable curves
and maps
Ian Morrison.................................................... 315
Riemann surfaces, integrable hierarchies, and singularity
theory
Yongbin Ruan.................................................. 371
SurveysinDifferentialGeometryXIV
Divisors in the moduli spaces of curves
Enrico Arbarello and Maurizio Cornalba
1. Introduction
ThecalculationbyHarer[12]ofthesecondhomologygroupsofthemod-
uli spaces of smooth curves over C can be regarded as a major step towards
the understanding of the enumerative geometry of the moduli spaces of
curves [21, 17]. However, from the point of view of an algebraic geome-
ter, Harer’s approach has the drawback of being entirely transcendental; in
addition, his proof is anything but simple. It would be desirable to provide
a proof of his result which is more elementary, and algebro-geometric in
nature. While this cannot be done at the moment, as we shall explain in
this note it is possible to reduce the transcendental part of the proof, at
least for homology with rational coefficients, to a single result, also due to
Harer[13],assertingthatthehomologyofM ,themodulispaceofsmooth
g,n
n-pointed genus g curves, vanishes above a certain explicit degree. A sketch
oftheproofofHarer’svanishingtheorem,whichisnotatalldifficult,willbe
presented in Section 5 of this survey. It must be observed that Harer’s van-
ishingresultisanimmediateconsequenceofanattractivealgebro-geometric
conjecture of Looijenga (Conjecture 1 in Section 5); an affirmative answer
to the conjecture would thus give a completely algebro-geometric proof of
Harer’stheoremonthesecondrationalhomologyofmodulispacesofcurves.
In this note we describe how one can calculate the first and second
rational (co)homology groups of M , and those of M , the moduli space
g,n g,n
of stable n-pointed curves of genus g, using only relatively simple algebraic
geometryandHarer’svanishingtheorem.ForM ,thisprogramwascarried
g,n
out in [5], where the third and fifth cohomology groups were also calculated
andshowntoalwaysvanish;inSection6,wegiveanoutlineoftheargument,
whichusesinanessentialwayasimpleHodge-theoreticresultduetoDeligne
[10]. In genus zero, we rely on Keel’s calculation of the Chow ring of M ;
0,n
a simple proof of Keel’s result in the case of divisors is presented in Section
4. We finally give a new proof of Harer’s theorem for H2(M ;Q); we also
g,n
Research partially supported by PRIN 2007 Spazi di moduli e teoria di Lie.
(cid:2)c2009InternationalPress
1
2 E.ARBARELLOANDM.CORNALBA
recover Mumford’s result asserting that H1(M ;Q) always vanishes for
g,n
g≥1. The idea is to use Deligne’s Gysin spectral sequence from [9], applied
to the pair consisting of M and its boundary∂M . This is possible since
g,n g,n
∂M is a divisor with normal crossings in M , if the latter is regarded as
g,n g,n
an orbifold. Roughly speaking, the Gysin spectral sequence calculates the
cohomology of the open variety M =M (cid:2)∂M in terms of the coho-
g,n g,n g,n
mology of the strata of the stratification of M by “multiple intersections”
g,n
of local components of ∂M . Knowing the first and second cohomology
g,n
groups of the completed moduli spaces M makes it possible to explicitly
g,n
compute the low terms of the spectral sequence, and to conclude. Knowing
the first and second homology of the moduli spaces of curves allows one to
also calculate the Picard groups of the latter, as done for instance in [4].
2. Boundary strata in M
g,n
As customary, we denote by M the moduli stack of stable n-pointed
g,n
genus g curves, and by M the corresponding coarse moduli space. It will
g,n
be notationally convenient to allow the marked points to be indexed by
an arbitrary set P with n=|P| elements, rather than by {1,...,n}. The
correspondingstackandspacewillbedenotedbyM andM .Ofcourse,
g,P g,P
we shall write M and M to indicate the open substack and subspace
g,P g,P
parametrizing smooth curves. By abuse of language, we shall usually view
M and M as complex orbifolds.
g,P g,P
As is well known, to any stable P-pointed curve C of genus g one may
attach a graph Γ, the so-called dual graph, as follows. The vertices of Γ are
the components of the normalization N of C, while the half-edges of Γ are
the points of N mapping to a node or to a marked point of C. The edges of
Γ are the pairs consisting of half-edges mapping to the same node, while the
half-edges coming from marked points are called legs. The vertices joined
by an edge {(cid:3),(cid:3)(cid:3)} are those which correspond to the components containing
(cid:3)
(cid:3) and (cid:3). The dual graph comes with two additional decorations; the legs
are labelled by P, and to each vertex v there is attached a non-negative
integer g , equal to the genus of the corresponding component of N. We
v
shall denote by V(Γ), X(Γ), E(Γ) the sets of vertices, half-edges, and edges
of Γ, respectively. The following formula holds:
(cid:2)
g=h1(Γ)+ g .
v
v∈V(Γ)
This implies, in particular, that g depends only on the combinatorial struc-
ture of Γ; we are thus justified in calling it the genus of Γ. The stability
condition for C is 2g−2+|P |>0, and hence can be stated purely in terms
of Γ. We shall say that Γ is a stable P-pointed graph of genus g. Given
(cid:3) (cid:3)
another P-pointed genus g graph Γ, an isomorphism between Γ and Γ con-
sists of bijections V(Γ)→V(Γ(cid:3)) and X(Γ)→X(Γ(cid:3)) respecting the graph