Poisson structures and birational morphisms associ

arXiv:alg-geom/9712022v119Dec1997POISSONSTRUCTURESANDBIRATIONALMORPHISMSASSOCIATEDWITHBUNDLESONELLIPTICCURVESA.POLISHCHUKLetXbeacomplexellipticcurve.Inthispaperwedefineanatu-ralPoissonstructureonthemodulispacesofstabletriples(E1,E2,Φ)whereE1,E2arealgebraicvectorbundlesonXoffixedranks(r1,r2)anddegrees(d1,d2),Φ:E2→E1isahomomorphism.Suchtriplesareconsidereduptoanisomorphismandthestabilityconditionde-pendsonarealparameterτ.ThesemodulispaceswereintroducedbyS.BradlowandO.Garcia-Prada[5].OurPoissonstructureinducesaPoissonstructureonsimilarmodulispaceswithfixeddeterminantsofE1andE2.ForE2=Oandsomevaluesofparameters(r1,r2,d1,d2,τ)thelattermodulispacesarejusttheprojectivespaces.Inparticular,oneofthesemodulispacesisPExt1(E,O),whereEisafixedstablebundle.ThecorrespondingPoissonstructuresonPExt1(E,O)werefirstdefinedandstudiedbyB.FeiginandA.Odesskii[7].Moreover,theyconstructedaflatfamilyofquadraticalgebras(Sklyaninalgebras)Qd,r(x)whered=degE,r=rkE,parametrizedbyx∈XsuchthatQd,r(0)isthesymmetricalgebraindvariablesandthequadraticPois-sonbracketonthesymmetricalgebraassociatedwiththisdeformationinducestheabovePoissonstructureonPd−1.ThealgebraQd,r(x)isdefinedastheassociativealgebraoverCwithdgeneratorsti,i∈Z/dZanddefiningrelationsXn∈Z/dZθj−i+(r−1)n(0)θj−i−n(−x)θrn(x)tr(j−n)tr(i+n)=0(0.1)fori,j∈Z/dZ,whereθm,m∈Z/dZarecertaintheta-functionsofleveldonX(see[6]).ForsomeothervaluesofparameterswegetasmodulispacetheprojectivespacePH0(E)whereEisastablebundleonX.Whentheparameterτchangescontinuoslythecorrespondingmodulispacedoesn’tchangeexceptwhenτpassesafinitenumberofrationalvalues,inwhichcasethemodulispaceundergoesbirationaltransformations1(flips)compatiblewithPoissonstructures.Inparticular,ifwestartwithastablebundleEofrankranddegreedrsuchthatdisrela-tivelyprimetor+1,thenwegetasequenceofPoissonbirationalmor-phismsconnectingPExt1(E,O)andPH0(E′)whereE′istheuniquestablebundleofrankr+1withdetE′≃detE.Ontheotherhand,usingFourier-MukaitransformoneconstructsanactionofacentralextensionofSL2(Z)byZonDb(X),thederivedcategoryofcoherentsheavesonX(see[9]).Usingthisactionwecon-structforeverystablebundleEofrankranddegreedanisomorphism(compatiblewithPoissonstructures)betweenthespacesPH0(E)andPExt1(E′,O)whereE′iscertainstablebundleofdegreedandrankr′satisfyingthecongruencerelationr·r′≡−1mod(d).AnotherapplicationofSL2(Z)-actiongivesaPoissonisomorphismbetweenPExt1(E,O)andPExt1(E′,O)whereEandE′arestablebundlesofthesamedegreedwhoseranksrandr′satisfythecongru-encer·r′≡1mod(d).Thisreflectsthefactnoticedin[6]thatthecorrespondingSklyaninalgebrasQd,r(x)andQd,r′(x)areisomorphicforanyx∈X.Itturnsoutthattheabovebirationalandregularisomorphismsofprojectivespacesfittogetherinthefollowingway.Foreveryd0wecanconsiderthedisjointunionof(d−1)-dimensionalprojectivespacesindexedbythesetofresiduesRd⊂Z/dZconsistingofrsuchthatbothrandr+1arerelativelyprimetod.Namely,foreveryr(0rd)thecorrespondingprojectivespaceisPExt1(E,O)whereEisstablebundleofrankranddegreed.ThentheabovebirationalandregularisomorphismsofprojectivespacesgeneratethebirationalactionofS3(thegroupofpermutationsin3letters)onthisdisjointunion.ThecorrespondingactionofS3onthesetofconnectedcomponentsRdisgeneratedbyoperatorsr7→r−1andr7→−r−1.Finally,weshowhowtogeneralizeourPoissonbracketstothesimilarmodulistacksforotherreductivegroups.Namely,wefixthefollowingdata:areductivegroupG,itsfinite-dimensionalrepresentationVandasymmetricg-invarianttensort∈(S2g)gwheregistheLiealgebraofG.Thesedatashouldsatisfythefollowingcondition:theoperatort∗:S2V→S2Vinducedbytshouldbezero.Thenweconsiderthemodulistackofpairs(P,s)wherePisaprincipalG-bundleonX,s∈V(P)isasectionoftheassociatedvectorbundle.GivenatrivializationofωXweconstructacanonicalPoissonstructureonthismodulistack.InthecasewhenG=GLr1×GLr2,Visthespaceofr1×r2-matrices,thereisanaturalchoiceofthetensortsuchthattheaboveconditionissatisfiedandwerecoverourPoissonstructureonmodulioftriples.ThesimplestcaseinvolvingothergroupsthanGLis2thecaseG=GSp(V)whereVisthesymplecticvectorspace,GSp(V)isthegroupofautomorphismspreservingthesymplecticformuptoanon-zeroconstant.Inthiscasethereisacanonicalchoiceoft,sowegetaPoissonstructureonthemodulistackofpairs(E,s)whereEisavectorbundleequippedwithasymplecticformE×E→L(whereLisalinebundle),sisasectionofE.1.StabletriplesLetusrecallthedefinitionofstabletriplesfrom[5].LetT=(E1,E2,Φ)beatripleconsistingoftwovectorbundlesE1andE2onXandahomomorphismΦ:E2→E1.Forarealparameterσtheσ-degreeofTisdefinedasfollows:degσ(T)=deg(E1)+deg(E2)+σ·rk(E2).Nowtheσ-slopeofTisdefinedbytheformulaμσ(T)=degσ(T)rk(E1)+rk(E2).NotethatifLisalinebundlethenwecandefineatensorofatripleTwithLnaturally,sothatonehasμσ(T⊗L)=μσ(T)+degL.ThetripleTiscalledσ-stableifforeverynon-zeropropersubtripleT′⊂Tonehasμσ(T′)μσ(T).Sometimesitisconvinienttointro-duceanotherstabilityparameterτ=μσ(T).ThecategoryoftriplesT=(E1,E2,Φ)isequivalenttothecategoryofextensions0→p∗E1→F→p∗E2(2)→0(1.1)onX×P1wherep:X×P1→Xistheprojection.Indeed,thespaceofsuchextensionsisExt1X×P1(p∗E2(2),p∗E1)≃HomX(E2,E1).ThisextensionhasauniqueSL2-equivariantstructureandassho