On moduli spaces of semistable sheaves on K3 surfaces [Elektronische Ressource] / vorgelegt von Markus Zowislok

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• Mathematics

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• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2010
• Nombre de lectures: 20

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MathematikOnvmoeduliMainzspacesJohannesofinsemistablewisloksheaPhvInformatikesboniK3vsurfacesMarkusDissertationinzurAprilErlangungysik,desundGradesderDoktorGutendererg-UnivNaturwissenscrshaftent?tamMainzForgelegtaconhZobgeb.ereicLahrhim082010-D77Datum28.05.2010derPr?fung:m?ndlichenfatherTyom(0;c; 0)
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rofai-VmoKdulihenspacesositivofdulr?umensemistabletlicsheahvGarbeseleonhproenjec-tgenerisctivBeispieleeZu-K3undsurfacestenotF?llecodiesenvheneredaubF?lleyauf[KLS06]die-Abstractthevisorencaseshenofom-MukcaibvulectorderselbcasesbtheoenenestigateRangvenasesonderewneuerellom-asbirationalmodesduliwspacesabgedecforonnongeneralGarbamplerodivisorshen-mitwitheregardnictoamplenthehinsicpderossibleonstruktionconstructiononofirreduziblenneweiten.examplesstelleofzucompactunirreducibleMosymplectic?umemerallgemeine-a-hnifolds.erwWkeaufestablishvavconnectionundtoontheonalreadyRang.inannvdieestigastoneirreduziblendeiten,mobdulionenspacesdulraumsorescgeneralisationsithereof,[KLS06]andktenwveMo-arehalbstabilerableentopextejektivnK3-Fl?cd-theF?lleinMuksoektordWedulr?umeknowiewniresultsMotozuallhofhentheDi-op-enhremaininghcasesm?glicforKrankneuer0vandkmanpaktenysymplektiscofMannigfaltigkthoseIforhpeinenositivsammenhangedenrank.ereitsIntersucpartictenular,dforrthesencasesVwrungeneencaneexcludeundtheeitereexistenceeofannnewErgebnisseexampleallesF?lleofoncompactenirreducibleomsymplec0tvieleivcGarbmanifoldsvlyingpbirationallyemoInsbvkerincompF?llenonenExistenztsBeiofpithevmokdulipaktenspace.symplektiscZusammenfassungMannigfaltigkIcdieh?unertersucomphetendieMonicliegen,hsgthlossenberden.ereitsdurc(H;A)
.Zusammenfassung4.1imoInS-equivtroonenduction.vshea1.State.of3the.art.1.1.1.Symplecticfamiliesvmoarietiesthe.16...es.......duli.......................abilit.....for.b..1.1.23.2Symplectic.resolutions......ten...............4.3...................4.6..2.1.3.Shea.vSymplecticesofon.K3.surfaces....duli.shea.3.1.w.spaces.........................4.for.es..3.1.4.General.ample.divisors....4.2.es.............ltration.......4.4...............4.55functor1.4.1.W.alls.for.t.w.o-dimensionalconstructionsheaduliv.es....39.......2.3.resolv.y.comp.ts....5.1.4.2.W.alls.for18one-dimensionalMosheaspacesvone-dimensionalesv.23.Morphisms.et.een.duli..............8.1.523SemistableResultsshea.v.es.on.K3.surfaces....................26.Mo.spacesAbstracttsiConshea.-semistable.v.29.Preliminaries11.1.6.Mo.duli.spaces.of.shea.v.es.on.K3.surfaces....29.Semistable.v................1.1.2.Irreducible.comp30onenJordan-H?ldertsand15alence2.1.A.decomp.osition....36.Flat.............................37.The.duli..........15.2.2.Pro.ducts.and.symme.t.ri38cTheproofductsmo.space...............iii.M (v)H;A
M (v)H
M (v)H;A
sM (v)H
......s....42.4.8.Lo.calwistedpropwe.rt.i60es.andresultsdimension.estimateson.s.....surface.81.On.......stable....46.4.9.Univ.ersal.families......y.........es.......59.discriminan.............Existence.v........47.55.7Mo.duli.spaces.for.t.w.o-dimensional.shea5.8v.es.49.5.1.Semistableabshea6.1vtes............6.2.shea.a.......Bibliograph.85.....5.5.the.t............49.5.2.The.mo.duli.space..5.6.ofofssheaProe-.construction..................63.More.on..................53.5.3.T....70.Results......erminalisations.for.The....4.7.CONTENTS.iv.....6.relation.t..72.The.to.wisted.t.ilit.75.T.stabili.y..................57.5.4.Birational.maps75bTeto-dimensionalwveenonmoK3duli.spaces..........76.y.Danksagung..X
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compacttheto.Bogomoloclassvcompactdecompknoositiont(see?hler[Bea83]nTheoremto2,simply[Ber55]Calabi-Yorw[Bog74]):areTheorem.canLmanifoldetadescribsolutionbexampleseTheawithcirreducibleompThereactholomorphicmanifoldtofsympleK?hlercompacttyp[Bea83]).edenedwit?herselytrivialpurerst?hlerChernbasedclass.conjecThenvtherbleeomplexiscompactaK?hlerniterst?talearecmanifoldsoveringuisamanifolds[Bea10]ofgeome-hw,advwhichcompactismanifolisomor-thephicypto([Huy99]thesymplecticprboanductypsuclofconstructureisTheyau78].result[YypyonbsymplectictrivialYisthewherure.eweryachcompactclassmanifoldsCherncistorus.aonlyprirreducibleojeconnectedctivemanifoldssimplytrivialcChernonnehencectecompactdsymplecticmanifoldandofadimensionmanifolds.rstisitsrecenifpublicationonlyonwithsymplectictrivitryaelanctoanonicertalThesheirreducibleafcticanddsandexactlyifirreducibleathRiccierkismanifoldseorypThetformyeasilybeK?hleronofirreducibledhforerkmanifolhomplexercbutcompactvAthereductiono,leaachexistencetroforInh,erkaremetrictoavirreducibleactmanifoldirroneau'sducofiCalabibletsympleUpcticnomanifoldonlyofeK?hlerfewtypofeirreduciandsymplecticedarewnaistheseisupadeformation:comp




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gateandositeselianvfactssheaconof..suitableByresolv[KLS06]OuritojisfromalmostAllcompletelyarieexcludedscthatvthis2constructionduliyicompelds(2.1.1)otherampleexamplesethanompthoseprmenO'Gradytionedanabforoevcollectsespaces-surfacesprop.osivttsiaones1.6.4othersummarisesvknobwnsurfacresults.HilbWoneofwirranbtistomorphisminexamplevsuitableestigateathewithmissingKummecasesoinbutpreonentsstODUCTIONrioctmooursemistableconsiderationsontotheirshealitvceseonstK3compsurfaces.theWhatsurfaceisvleftresulthererelationaretsthedulimostableduliPropspacesetofasectivemistablehemesheascvE.g.esorwithvrankstabeduli,constructedgivducibleeofnersttherChernbirclass,eandabEulerO'Gradycaharacteristicforpacc,exampleandeliantheciatedmoidulivspacessurfacesonduliandmoethedistincttocof,sheaHilbvviesknwithwnMukonaidulivofectorsheaphicesthatK3areandsemistablesymplecticwithabirespyeIncthaptertowainnoteri-generalirreducibleampleonendivisorofisomomo.spaceInK3cerhapter.1rstwgivearecalltotheonenstateofofmothespacesart.tainingWsheaees:givosition.eLtheodenitioneofpraesymplecticK3ve,arietanydivisor,andertcitethethsurfaces.eabimpK3iseswithshearesultlofsemiHuybrecspaceshmotsandthatanbirationaleprocjectivonenteeirreduciblecansymplvecticThenmanifoldseareadeformationationalequivojalenctivet.oWexamplese[O'G03]recallofthe6-dimensionaldeniti[O'G99]onofofathedeMukompaiionv10-dimensionalectorsurfaceofabatosheafassoonesatK3andsurfacertogethergeneralisedwithK3rstaproptsertiespandadiscusschoictheofnotionairwiseofirraduciblegeneralompampleofdhemesiertonranksurfacesK3visor.forTheINTRlastsectionortan.t