Grassmannians, moduli spaces and vector bundles : Clay Mathematics Institute Workshop Moduli Spaces and Vector Bundles, with a view towards coherent sheaves, October 6-11, 2006, Cambridge, Massachusetts

190 Pages·2011·1.89 MB·English

by  David A. Ellwood

Clay Mathematics Proceedings
Volume 14
Grassmannians, Moduli Spaces
and Vector Bundles
Clay Mathematics Institute Workshop
Moduli Spaces of Vector Bundles,
with a View Towards Coherent Sheaves
October 6–11, 2006
Cambridge, Massachusetts
Clay Mathematics Institute, Cambridge, MA P. E. Newstead
David A. Ellwood
Emma Previato
Editors
American Mathematical Society
Clay Mathematics Institute
cmip-14-previato2-cov.indd 1 4/11/11 12:55 PM
Grassmannians, Moduli Spaces
and Vector Bundles

Clay Mathematics Proceedings
Volume 14
Grassmannians, Moduli Spaces
and Vector Bundles
Clay Mathematics Institute Workshop
Moduli Spaces of Vector Bundles,
with a View Towards Coherent Sheaves
October 6–11, 2006
Cambridge, Massachusetts

David A. Ellwood
Emma Previato
Editors
American Mathematical Society
Clay Mathematics Institute
2000 Mathematics Subject Classification. Primary 13D03, 14D20, 14F22, 14H60, 14K10,
22E57, 47B35, 53B10,53D30, 57R56.
Coverphoto of PeterNewstead courtesy of Dr. CristianGonzalez-Martinez.
Library of Congress Cataloging-in-Publication Data
ClayMathematicsInstituteWorkshoponModuliSpacesofVectorBundles,withaViewTowards
CoherentSheaves(2006: Cambridge,Mass.)
Grassmannians,modulispacesandvectorbundles: ClayMathematicsInstituteWorkshopon
Moduli Spaces of Vector Bundles, with a View Towards Coherent Sheaves (2006 : Cambridge,
Massachusetts,October6–11,2006/DavidA.Ellwood,EmmaPreviato,Editors.
p.cm. —(Claymathematicsproceedings;v.14)
Includesbibliographicalreferences.
ISBN978-0-8218-5205-7(alk.paper)
1. Commutative rings—Congresses. 2. Moduli theory—Congresses. 3. Vector bundles—
Congresses. I.Ellwood,D.(David),1966– II.Previato,Emma. III.Title.
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Contents
Introduction vii
Hitchin’sProjectivelyFlatConnection,ToeplitzOperatorsandtheAsymptotic
Expansion of TQFT Curve Operators 1
Jørgen Ellegaard Andersen and Niels Leth Gammelgaard
Koszul Cohomology and Applications to Moduli 25
Marian Aprodu and Gavril Farkas
Reider’s Theorem and Thaddeus Pairs Revisited 51
Daniele Arcara and Aaron Bertram
Intersection Pairings in Singular Moduli Spaces of Bundles 69
Lisa Jeffrey
The Beilinson-Drinfeld Grassmannian and Symplectic Knot Homology 81
Joel Kamnitzer
Arithmetic Aspects of Moduli Spaces of Sheaves on Curves 95
Max Lieblich
Existence of α-Stable Coherent Systems on Algebraic Curves 121
P. E. Newstead
Regularity on Abelian Varieties III: Relationship with Generic Vanishing and
Applications 141
Giuseppe Pareschi and Mihnea Popa
Vector Bundles on Reducible Curves and Applications 169
Montserrat Teixidor i Bigas
v

Introduction
In2006,PeterE.NewsteadpaidhisfirstacademicvisittoNorthAmericaafter
the1960s,andtheoccasionoriginatedanumberofworkshopsandconferencesinhis
honor. The editors of this volume, together with Montserrat Teixidor i Bigas, or-
ganizedaClayMathematicsInstituteworkshop, “Moduli spaces ofvectorbundles,
with a view toward coherent sheaves” (October 6-10, 2006). The experts convened
produced a vigorous confluence of so many different techniques and discussed such
deep connections that we felt a proceedings volume would be a valuable asset to
thecommunityofmathematiciansandphysicists; whenaparticipantwasnotavail-
able to write for this volume, state-of-the-art coverage of the topic was provided
through the generosity of an alternative expert.
PeterE.NewsteadearnedhisPh.D.fromtheUniversityofCambridgein1966;
both John A. Todd and Michael F. Atiyah supervised his doctoral work. From the
beginning of his career, he was interested in topological properties of classification
spaces of vector bundles. Geometric Invariant Theory was re-invigorated around
that time by the refinement of concepts of (semi-)stability and projective models,
and Newstead contributed some of the more original and deeper constructions,
for example the projective models for rank-2 bundles of fixed determinant of odd
degree over a curve of genus two, a quadratic complex (obtained also, using a
different method, by M.S. Narasimhan with S. Ramanan), and a topological proof
for non-existence of universal bundles in certain cases. He played a prominent
role in the development of a major area by focusing on moduli of vector bundles
over an algebraic curve, was a main contributor to a Brill-Noether geography for
these spaces, and his topological results led him to make major contributions (cf.
[N1-2] and [KN]) to the description, by generators and relations, of the rational
cohomologyalgebraH∗(SU (2,L))for themoduli spaceSU (2,L)ofstable rank-
X X
2bundlesoverX withfixeddeterminantL,whereX isacompactRiemannsurface
of genus g ≥ 2, and L a line bundle of odd degree over X (the higher-rank case
was then settled in [EK]). The main concerns of his current work are coherent
systemsonalgebraiccurvesandPicardbundles. Thisbird’s-eyeviewofNewstead’s
work omits several topics, ranging from invariants of group action to algebraic
geometryoverthereals,conicbundlesandotherspecialprojectivevarietiesdefined
by quadrics, and compactifications of moduli spaces, and is merely intended to
serve as orientation for the readers of the present volume.
Thisvolumeofcutting-edgecontributionsprovidesacollectionofproblemsand
methods that are greatly enriching our understanding of moduli spaces and their
applications. It should be accessible to non-experts, as well as further the interac-
tion among researchers specializing in various aspects of these spaces. Indeed, we
vii
viii INTRODUCTION
hope this volume will impress the reader with the diversity of ideas and techniques
that are brought together by the nature of these varieties.
In brief and non-technical terms, the volume covers the following areas. An
aspect of moduli spaces that recently emerged is the disparate set of dualities that
parallel the classical Hecke correspondence of number theory. In modern terms,
suchapairingoftwovariables(ortwocategories)isaFourier-Mukai-Laumontrans-
form; implications go under the heading of geometric Langlands program, the area
of Kamnitzer’s article. Pareschi and Popa offer original techniques in the derived-
theoretic study of regularity and generic vanishing for coherent sheaves on abelian
varieties, with applications to the study of vector bundles, as well as that of lin-
ear series on irregular varieties. Also on the theme of moduli spaces, Aprodu and
Farkasontheonehand,Jeffreyontheother,providetechniquestoanalyzedifferent
properties: respectively, applications of Koszul cohomology to the study of various
modulispaces, andsymplectic-geometricmethodsforintersectioncohomologyover
singular moduli spaces. Teixidor treats moduli spaces of vector bundles over re-
ducible curves, a delicate issue with promising applications to integrable systems.
Arcara and Bertram work torwards a concept of stability for bundles over surfaces
and conjecturally over threefolds. Using the Brauer group, Lieblich relates the
geometry of moduli spaces to the properties of certain non-commutative algebras
and to arithmetic local-to-global principles. Andersen and Gammelgaard’s paper
addresses a quantization of the moduli space: the fibration of the moduli space of
curves given by the moduli of bundles admits a projective connection whose as-
sociated operator generalizes the heat equation – it was defined independently by
N. Hitchin, and by S. Axelrod with S. Della Pietra and E. Witten. Finally, the
workshop’s guest of honor, Peter Newstead himself, offers a cutting-edge overview
of his current area of work, coherent systems over algebraic curves; may we salute
it as the Brill-Noether theory of the XXI century?
We hope you enjoy the book and find it as inspiring as we do.
David A. Ellwood, Cambridge
Emma Previato, Boston
References
[EK] R.EarlandF.Kirwan,Completesetsofrelationsinthecohomologyringsofmodulispaces
of holomorphic bundles and parabolic bundles over a Riemannsurface, Proc. London Math.
Soc. (3)89(2004),no.3,570–622.
[KN] A.D.KingandP.E.Newstead,Onthecohomologyringofthemodulispaceofrank2vector
bundlesonacurve,Topology37(1998),no.2,407–418.
[N1] P.E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve,
Trans. Amer. Math. Soc.169(1972),337–345.
[N2] P.E. Newstead, On the relations between characteristic classes of stable bundles of rank 2
overanalgebraiccurve,Bull. Amer. Math. Soc. (N.S.)10(1984),no.2,292–294.
ClayMathematicsProceedings
Volume14,2011
Hitchin’s Projectively Flat Connection, Toeplitz Operators
and the Asymptotic Expansion of TQFT Curve Operators
Jørgen Ellegaard Andersen and Niels Leth Gammelgaard
Abstract. Inthispaper,wewillprovideareviewofthegeometricconstruc-
tion,proposedbyWitten,oftheSU(n)quantumrepresentationsofthemap-
ping class groups which are part of the Reshetikhin-Turaev TQFT for the
quantumgroupUq(sl(n,C)). Inparticular,werecallthedifferentialgeometric
construction of Hitchin’s projectively flat connection in the bundleover
Teichmu¨ller space obtained by push-forward of the determinant line bundle
overthemodulispaceofrankn,fixeddeterminant,semi-stablebundlesfiber-
ing over Teichmu¨ller space. We recall the relation between the Hitchin con-
nectionandToeplitzoperatorswhichwasfirstusedbythefirstnamedauthor
toprovetheasymptoticfaithfulnessoftheSU(n)quantumrepresentationsof
the mapping class groups. We further review the construction of the formal
Hitchin connection, and we discuss its relation to the full asymptotic expan-
sion of the curve operators of Topological Quantum Field Theory. We then
goontoidentifythefirsttermsintheformalparalleltransportoftheHitchin
connectionexplicitly. Thisallowsustoidentifythefirsttermsintheresulting
starproductonfunctionsonthemodulispace. Thisisseentoagreewiththe
first term in the star-product on holonomy functions on these moduli spaces
definedbyAndersen,MattesandReshetikhin.
1. Introduction
Witten constructed, via path integral techniques, a quantization of Chern-
Simons theory in 2+1 dimensions, and he argued in [Wi] that this produced a
TQFT,indexedbyacompactsimpleLiegroupandanintegerlevelk. Forthegroup
SU(n)andlevelk,letusdenotethisTQFTbyZ(n). Wittenarguesin[Wi]thatthe
k
theoryZ(2) determinestheJonespolynomialofaknotinS3. Combinatorially, this
k
theory was first constructed by Reshetikhin and Turaev, using the representation
theory of U (sl(n,C)) at q = e(2πi)/(k+n), in [RT1] and [RT2]. Subsequently, the
q
TQFT’sZ(n)wereconstructedusingskeintheorybyBlanchet,Habegger,Masbaum
k
and Vogel in [BHMV1], [BHMV2] and [B1].
Thetwo-dimensionalpartoftheTQFTZ(n)isamodularfunctorwithacertain
k
label set. For this TQFT, the label set Λ(n) is a finite subset (depending on k)
k
of the set of finite dimensional irreducible representations of SU(n). We use the
2010MathematicsSubjectClassification. Primary53D50;Secondary47D35,53D57,57R56.
(cid:2)c 2011JørgenEllegaardAndersenandNielsLethGammelgaard
1