The Euler number of O'Gradys 10-dimensional symplectic manifold [Elektronische Ressource] / Sergiy Mozgovyy

Description

MathematikThevyyEulerbndenumderbnerNikoPhfInformatiO'Grady'sG20.10.2006ersit?t-dimensionalSergiysymplecticorenmanifoldUkraineDissertationhzurysik,ErlangungunddeskGradesJohannesDoktorutendererg-UnivNaturwissenscihaftenMainzamMozgoFgebacinholaev,bMainz,ereic10AprilTPr?fung:aghender20.m2007?ndlicbAbstractsemistableWMoeplecomputeinthebEulerrvntoumseber,ertheofmotheonspacepr-dimensionaldissertationexceptionaltheirreduciblemosymplecticindepmanifoldEulerconstructedLagrangianbrestyers.O'Gradyare.spacesThevideasingularisandtoncconstructofitsdevLagrangiancalculationbrationnandoftospaces.computearethendenEulerKeywnumumsymplecticberserstheofofthebbThoseers.ersIttheturnsdulioutofthatsheaalmostesalltheofcutheesbtheersihaivpartetheEulerisnotedumtheboferEulerduliumanderstherthoseeforedulitheTheproblemresultsisofreducedetottheterest.computationords:ofnthebEulerIrreduciblenmanifold,umbration,bof100vZusammenfassungbstabilenWirEulerzahlenbhnereceit,hnensedierEulerzahlDiesederEulerdulraumBerec-dimensionalenasern.exzdulr?umeeptionellsingul?reneistndieserirredu-aziblenteresse.symplektiscblehenaufMannigfaltigkungeit,?brigendieFvsindononO'Gradyenken.onstruiertdieserwurde.

Sujets

• Mathematics

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2007
• Nombre de lectures: 21
• Langue: English

Extrait

10
InformatiEulerNiknnumPhbGervyyodenfMathematikO'Grady'sderThebersit?t-dimensionalSergiysymplecticorenmanifoldUkraineDissertationhzurysik,ErlangungunddeskGradesJohannesDoktorutendererg-UnivNaturwissenscihaftenMainzamMozgoFgebacinholaev,bMainz,er20.10.2006eicAprilTPr?fung:aghender20.m2007?ndlic10
0
moevcomputeresttheofEulerEulernareumandbcalculationerindepofLagrangiantheers.Wspaces-dimensionalsingularexceptionalncirreducibledevsymplecticbmanifoldseconstructedinber,yspaceO'Gradythe.bThemoideasemistableisontorvconstructpritspleLagrangiandissertationbrationtoandthetoofcomputespaces.theareEulerndennKeywumumbsymplecticersMoofthetheofbbers.ThoseItersturnstheoutdulithatofalmostsheaallesofthethecubesersthehaiviepartEulerthenisumotedbtheerofAbstractEulernumanderstherthoseeforedulitheTheproblemresultsisofreducedetottheterest.computationords:ofnthebEulerIrreduciblenmanifold,umbration,bduliersof10
0
teresse.benerec?brigenhnenderdieaufEulerzahlsindderdieserWirsul-dimensionalenbleexzungeptionellFeonnen.irredu-rziblendulr?umesymplektiscehenEulerMannigfaltigkeit,eit,BerecdieEulerzahlenvasern.onseO'Gradydulr?umekbstabilenonstruiertsingul?renwurde.rDieistIdeehnbdieserestehDiesetadarin,unabh?ngigemzun?chlagwhstIrreduzeineheLagrangefaserungMozudiekhnonstruierenderundderdannFdieDieseEulerzahlenaderrnFMoasernvzuhalbGarberecaufhnen.KurvEsDestelltHauptteilsicDissertationhdeheraus,BerecdassungfastEulerzahlenalleMoFgewidmet.asernRedietEulerzahltZusammenfassungsindvonhabInen,Scund?rter:deswZahl,egenireduziertsymplektiscsicMannigfaltigkhLagrangefaserung,dasdulraumProblem-M (4n+2) |2H|X
M (4n+2)X
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sM1
sM2
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n541.2.Fibrations.ofvirreduc.ibl.e4.3.1symplecticermanifoldsdouble.......shea.Mo.......427.1.33.2O'Grady's.exampleerand.itserbirational.v4.2ersion..um...1.........duli.er.es8.1.4.Fibration..bien.....Euler.duli.....Euler.sextics.............4.3.strata.Basic...um.....62.of.....Euler.....314of2oMocurvduliSheaspacesvofesshea.v.es.o.v.erinreducedspacecurv.es.17.2.1.T46ensorumproaducts.with.line.bundles....of.um.4.1...............E-function.es.of....17.2.2.MoEulerduliersspacesspaceso5v.er.non-rational.curvEulereser.duction.........The.b.In.........4.3.319um2.3tenSemistable.shea.v.es39onMoaspacesrationalsheacurvesevwithdoubleoneesno3.1dev.o.er.curv....20.2.3.1.Se.m.istable.shea.v.es..42.Extensions.am.t.....................3.3.n.b.of.mo.space2.1.2.3.2.Se.m.istable.c.hains.and52cyclesComputation.the.n.b.54.General....................24.2.3.3.Classication.of54semistableStringysheaofv.esv.semistable.................61.The33n2.4bMoofduliofspacesdulio1.1vconstructionser.a.rational.curv.e.with62manTheynnobdesof.1........34.2.5.One.sp.ecial.c4.3.2aseEuler.um.er.tro.ts..............iii..63.The.n.b.of.Con..................63..erAErkl?rungGrothendiecnkEulersp80ectral84sequenceson65umBBibliographDescriptioncofwledgementorsionvitaefreeivsheathevnesb6976CySomeAfactskaboouttsatCurriculumfamilies867487DPreliminariesX
0 2H (X,Ω )X
b = 222
[n]X X n ≥ 2
b = 232
nK X X
n> 2 b = 72
b = 242
b = 82
X
X Y
n m −24e(n)q = (1−q ) ,
n≥0 m≥1
[n]e(n) :=e(X ) e(2) = 324 e(5) = 176256
simrnherebisareclass.andChernbrstdeterminedtrivialK?hler.of2.vTheandHilbWertertscsecondhemesahholomorphictanwiabmanifoldstheir,BettiwhereHilbK?hlerarietiesis[18,aorK3-surfaceofandacompactformthe1.ofmanifolds,.thatHcalledehasreifstonesthebuildingmanifoldsthevareparticular.erse3.ts.GeneuralrsizescdKummerKummererevG?ttscarietiesusingmanifolds,conjectures.autheCalabi-Yb,HiwherehemesandumisyantK3-surfaces.wwn,o-dimensionaltotorussucand.toriacomplexfoldwithform.;Hereconnectedalong,manifolds,symplecticcticisestudy.the4.listed10-dimensionaloexampleeinintrondingduceddivbinyarianO'GradyThe[36].nHeremsympleholomorphicofirreducibleertThehemes.generalized5.v6-dimensionalwexamplecomputedinytroheduced19]btheyeilO'GradyF[37].example,HereEulerductionumeerstrotheInlbandscergelofusingK3-surfacedecomparetheorembBeilinson-Bernstein-theerulapBettibThe.Thecomputationnamely:theknondeformation,buptogetherhthefewdgeOnlyumRecallerswhereascompactbmani-G?ttscistogeneratingdesymplecticfaormation.e.g.AnotherandproblemplywisorthitspmanifoldecialirreducibleattentionerAnothersheaofesBetti5]umitserstwithmixedHodgenthebirre-wduciblemadesymplecticymanifoldsheupSo.[20],IttheisositionanofinDeligne-Gabbtriguingforproblemervtoseprovv[eandthatrenementhisforlistHoexhausts1allX
H
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2H = 2 H M (2,0,4) =X
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M (0,2H,2)X
M (4n + 2)X
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M (0,2H,2) =M (4n+2)X X
-M (4n+2) |2H|X
C ∈|2H| M (4n+2)C
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|2H|
M (4n+2)X
e(5)+2e(2) = 176904
corresppart)eandthethenofonewssho10-dimensionalwspthatumitman-hasouratheresolutionduliofasingularitiesinducedsuccurvhdeterminethatespthe2symplecticbftheormideacanstrata)btheeofextendedthetodthespacewholeoutresolutionforandTremainsexamplenon-degenerate.theSucaccordinghnumbainresolutiontheseisthecalledsymplecticaordesym-bplecticletresolution.TItbisthethebrequiredsheaf10-dimensionalsuppirreduciblevsymplecticaimmanifold.umDeformingRapagnettathereptheolarizedrespK3-surfaces,O'Gronencanbassumeumthatbnonsingularmputethetheertsvc(obformandsymplecticbandTheoremaady'swithqualedtoequippi,w-upswheretoenisofansoamplegoal.ditovisorn(seeof[35]).recallThenthebexplaincan(andspacenduliofmoeThisxa.ofclassnChernmapssecondtheandtoclassTheChernbrationrsta,towhere[39].inEulertheerlastdulimo[37]duli,spaceHilbwialemoconsideconsideredrtotheolarizationHilbItertthepbolynomialbrewithnonzerorespniteectertointheerepcolarizationEulerwithers.ofItthewwassucproes,vnedofbbythemO'Gradytheirinwith[35,toPTheaofrsympletis2.2]arianthat[40].theredulesisccurringanbirationalblomapallobusetcomputewEulereenumoneresthevreshealution,semistableultimaterankInandrofndspaceEulerduliummoerar)con-lbrieyuus(singcomputationaofconstructtheandoolarizationmple.palsogeneralEulertlyumsucieners.itsThewlastconstructmobrationduliespaceO'Grady'scaneralsoumbEulerethateaquippinedmowithspaceaitssymplecticort.formbre(othisvoererthecurvnonsingularcomputepart)isandOurithasalsosamehasnabsymplecticasresolution,mosospacebandothysymplecticaresolutionswhearethebirationalertandolynomhenceindieomorphiclastbduliyisawith.ectoftheHuybrecphonts.[25].turnsIthattEulerfolloumwserthatthebcanothesymplecticonlyresolutionsahanvbeoftheessameyEulercomputedn.umoboer.theThensymbpleofcticstrataresolutionwof6-dimensionaltheofmo,duliespaceallwithhK3-surfaceurvecomputejectivEulerproumaersistheobtainedondingbersysumcertainupblotow-ups.memComputationershipofrtheectEulerstrata.n1.umEulerberersO'grof10-dimensionalthecticstrataifoldofeconsidertoevWTheexample.Saitothisdueofmostructionotheresult[5]M (4n+1) e(X ) =X
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-M (4n + 2) |2H|X
M (4n+2) C ∈|2H|X
M (4n+2) 1 2C
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k[x,y]/(xy)
devFirstesmoondulisucinhes.curspacevtationes.wLetofusfnovwhasdescribesecomptheThestrucnamelyture[14],ofthetheUsingthesisdevinvmoreesdetail.theThevrstncgreatlyhapterAfterisypdetervofotedistouctheosabledenitiontoofvthereO'Gradytheexamplosableetheandumdescribingtoitsspacesprop(perties.theFirst,vwnonzeroeofrecalletheerbasiconenresultsofconcerningresuthebmovdulistudyspaceserofcurvsemistablewsheacoursevnesequalsonthattheeproejectivweclassives.arieties.esThenofwtheehinbinato-trobduceBurbanthewithframe-[9]wesorkaiofclassication(rational)erLagrangiancomparison,brationstforstudyamomoresemistableconceptualonexpreducible)ositioneofdulithethosefurthethatrthematerialonofonenthecurvcallhapter.lWthatecurvrecallnonrationalthetheconstructionbomofzero.O'Grady'st10-dimensionalnsymplecticofveariettoyw(thehesymplecticoresolutiontofofsemistablewithoftsspaceswillduliinmothethetheofb,InwhereererscurvisearationalK3-surface)oneandHerealsoactuallygivmoreeneeitswbirationalallvsheaerapproacsiuseonc(wtheandsymplectictheyresolutionosableofonbcurvumofndata.Eulertthecomparedoforkcomputationwhothebundles),wwhicandhclassiedissheamoreranksuitableFfortheyourtocomputationsniteasoitringpEulerossessesJusttheoLagrangianedbrationthemenotionedtheabdulioofvsheae.esWreducedeossiblyconstructcurvthisWbrationstudyinmothespacnalofsection.sheaInestuhaietivsameelyrank,allitcompistsathemape.ofofconsistswswilsiprotheethewhenevofacoreeTheareducible.compthemt,sendingEuleraumsheafertotheitdulisuppequalsort.ThisWlereduceshathevumeertriedcurvtowbhaeequiteconsider.accuratethatproevingtthatmoitspacesisvadierenregulartmorphismesofreducedsceshemes.rationalTheonenbthatereofencounoflatersometheandofreducedcompu-nonofhemEulertumoervtheer.aofcurv.erstofesomewes,studycurvaexoticcurvquitewithisnorelated.towthedomomdulihspacethanterencound,e,willeefywindecomp.semistableAllvsheaThevheseingothebalastkmotheduliorkspaceDrozdhaGreuelvwhereeclassiedranksindecompcomputationssheaoresoursuc.aTheeoftermssheacertaincomtherialsolvOurbsulGelfandcanPe[withOurwhofcertainly[7],elemenclassiedthestableEulerandntheumorkbBurbanersKreu?lerofwhotheindecompmosemistabledulivspacesofofzero.rankthecourseourier-Muksemistabletransform,sheareducedvproblemestheonofvdimensionalariousdulescurvvthees.bThensecondthecforhapter3isleitmotif,ofproblemtheedsecondyandandthirdonomarevc17].haptersapproacisisthemorecalculationtaryofn > 1
2
2
n
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n
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2I C C I = 0 I Cred red
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OCred
Cred
Cred
M (4n + 2)X
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-π : X P B X
B PicX =Z[H] H
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P B |2H|
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C B0
C X
C
M (4n+2)X
rbey[16].Wtheue[42].erythingHefourthhaseshoinwncomputethatisitngiswequaloftoTesextic..IninthehacasemofwtheumeveensheadegreeJacobianwerecalculationwiofllandshosingularitieswspthatntheaEulerfornwhereumtlybitsermequalshvtforofanesychaduli.umInoourcorrespcal