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. QA251.3.C54 2011 514(cid:2).224—dc22 2011001894 Copying and reprinting. 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VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 161514131211 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