A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

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aAFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACESC.Bartocci, U.Bruzzo, andD.Hern´andezRuip´erez¶ DipartimentodiMatematica,Universit`adiGenova,Italia ScuolaInternazionaleSuperiorediStudiAvanzati(SISSA—ISAS),Trieste,Italia¶DepartamentodeMatem´aticaPurayAplicada,UniversidaddeSalamanca,Espa naRevised—6August1996Abstract.Wede neaFourier-MukaitransformforsheavesonK3surfacesoverC,andshowthatitmapspolystablebundlestopolystableones.Ther oleofvarietytothegivenK3surfaceXishereplayedbyasuitablecomponentmodulispaceofstablesheavesonX.ForawideclassofK3surfacesXtobeisomorphictoX;thentheFourier-Mukaitransformisinvertible, Xcanbeandthe “dual”ofthechosenimageofazero-degreestablebundleFisstableandhasthesameEulercharacteristicasF.1.IntroductionandpreliminariesMukai’sfunctormaybede nedwithinafairlygeneralsetting;giventwoschemes

X,Y(of nitetypeoveranalgebraicallyclosed eldk),andanelementQinthederivedcategoryD(X×Y)ofOX×Y-modules,Mukai[18]de nedafunctorSX→YfromthederivedcategoryD (X)toD (Y),

SL

X→Y(E)=Rπ (Q π E)

(hereπ:X×Y→Xandπ :X×Y→Yarethenaturalprojections).Mukaihasprovedthat,whenXisanabelianvariety,

ebundleonX×X

ofcategories.IfoneisinterestedintransformingX→sheavesX givesrise itsdualvariety,andQthePoincar´ Y=X

,thefunctorStoanequivalence

ratherthancomplexes

onecanintroduce(followingMukai)thenotionofWITisheaves:anOX-moduleE

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

2´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

issaidtobeWITiifitsFourier-Mukaitransformisconcentratedindegreei,i.e.HRπ (Q π E)=0forallk=i.

isitsdual.IfFisaµ-stablevectorLetXisbeapolarizedabeliansurfaceandX bundleofrank≥2andzerodegree,thenitisWIT1,anditsMukaitransformF )ofdegreeisagainaµ-stablebundle(withrespecttoasuitablepolarizationonX

zero.ThisresultwasprovedbyFahlaouiandLaszlo[10]andMaciocia[14],albeitSchenk[21]andBraamandvanBaal[7]hadpreviouslyobtainedacompletelyequivalentresultinthedi erential-geometricsetting:theNahm-Fouriertransformofaninstantonona atfour-torus(withno atfactors)isaninstantonoverthedualtorus(cf.also[9],and,foradetailedproofoftheequivalenceofthetwoapproaches,

[3]).Thekeyremarkwhichmakesitpossibletorelatethealgebraic-geometrictreatmenttothedi erential-geometriconeisthattheFourier-Mukaitransformisinterpretableastheindexofasuitablefamilyofellipticoperatorsparametrizedby ,verymuchinthespiritofGrothendieck-Illusie’sapproachtothetargetspaceX

thede nitionofindexofarelativeellipticcomplex.

TheFourier-Mukaitransformcanbestudiedalsointhecaseofnonabelianvari-eties.ThegeneralideaistoconsideravarietyX,somemodulispaceYof‘geometricobjects’overX,and(ifpossible)a‘universalsheaf’QovertheproductX×Y.InthepresentpaperweconsiderthecaseofasmoothalgebraicK3surfaceXoverCwitha xedamplelinebundleHonit;ther oleofdualvarietyishereplayedbythemodulispaceY=MH(v)ofGieseker-stablesheaves(withrespecttoH)EonXhavinga xedv,where

v(E)=ch(E) kL

=MH(v)isaForawidefamilyofK3surfacesX,wecanchoosevsothatX

K3surfaceisomorphictoX.ThenonemptinessofMH(v)inthiscasehasbeenprovedin[4]bydirectmethods.ThenweprovethattheFourier-Mukaitransformisinvertible,justasinthecaseofabeliansurfaces:

Theorem2.LetFbeanWITisheafonX.ThenitsFourier-Mukaitransform =Riπ ,whoseFourier-MukaitransformF (π F Q)isaWIT2 isheafonX Q )isisomorphictoF.R2 iπ ( π F

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

AFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACES3Weendthissectionby xingsometerminology.LetXandYbecompactcomplexmanifolds,andletQbea xedcoherentsheafonX×Y, atoverOY.Letπ,π betheprojectionsontothetwofactorsofX×Y.

De nition1.AsheafEonXsatis estheithWeakIndexTheoremcondition(i.e.itisWITi)ifRjπ (π E Q)=0forj=i;similarly,thesheafEissaidtosatisfytheithIndexTheoremcondition(i.e.itisITi)ifHj( π 1(y),E Q|p 1(y))=0forj=iandally∈Y.

ThebasechangetheoremimpliesthatthesheafEisITiifandonlyifitisbothWITiandRiπ (π E Q)islocallyfree.

IfeitherQis atoverOX×Y,orFis atoverOX,thesheavesRkπ (π F Q)arethecohomologysheavesoftheFourier-MukaitransformRπ (π F Q)inthe =Riπderivedcategory.Then,ifFisWITiwecallthesheafF (π F Q)onYitsFourier-Mukaitransform.

2.Fourier-MukaitransformonK3surfaces

2.1.Hyperk¨ahlermanifoldsandquaternionicinstantons.Ahyperk¨ahlermanifoldisa4n-dimensionalRiemannianmanifoldXwhichadmitsthreecom-plexstructuresI,JandK,compatiblewiththeRiemannianstructure,suchthatIJ=K.Onahyperk¨ahlermanifoldonecanintroduceageneralizednotionofinstanton[15].ThethreeendomorphismsI,J,KofTX Callowonetode neanendomorphismφofΛ2T X C,

φ=I I+J J+K K.

Thissatis esφ2=2φ+3,sothatateveryx∈Xonehasaneigenspacedecompo-sition

(1) Λ2(TxX C)=V1⊕V2L

correspondingtotheeigenvalues3and 1ofφ,respectively.IfEisaC∞complexvectorbundleonX,withconnection andcurvatureR ,wesaythatthepair(E, )isaquaternionicinstantonifR ,regardedasasectionofEnd(E) Λ2T X,hasnocomponentinV2.IfXhasdimension4thisagreeswiththeusualde nitionofinstanton,sinceinthatcasethesplitting(1)isnomorethanthedecompositionofthespaceoftwo-formsintoselfdualandanti-selfdualforms.Itisquiteevidentthat(E, )isaquaternionicinstantonifandonlyifthecurvatureR isoftype(1,1)withrespecttoallthecomplexstructuresofXcompatiblewithitshyperk¨ahlerstructure.Moreover,(E, )isanEinstein-HermitebundlewithrespecttoalltheinducedK¨ahlerstructures;thus,foranycompatiblecomplexstructureonX,thebundleEadmitsaholomorphicstructure,andthesheafofitsholomorphicsectionisthenµ-polystablewithrespecttoapolarizationgivenbytheK¨ahlerformdeterminedbythegivencomplexstructure(werecallthatacoherentsheafissaidtobeµ-polystableifitisadirectsumofstablesheaveshavingthesameslope).

2.2.Assumptionsonthemodulispace.LetXbeaprojectiveK3surfaceoverC.Thecupproductde nesaZ-valuedpairingonthegradedringH (X,Z):

′′′′′′

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

4´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

Forevery sheafFonXwede netheMukaivectorv(F)∈H(X,Z)astheelement

ch(F)

Nowwe xaMukaivectorv=(r, ,s)satisfyingtheassumptionA1andthefollowingadditionalassumptions:

A2.Thedivisor hasdegreezeroandr>1;

A3.ThemodulispaceMH(v)isnotempty,andparametrizesµ-stablesheaves.

InthissituationtheuniversalsheafQonX×MH(v)islocallyfreebyvirtueof

[20],Corollary3.10,thatwerecallinthefollowingform:

Proposition2.Ifv=(r, ,s)isisotropicandr>1,everyµ-stablesheafFonXwithv(F)=vislocallyfree.

1Weadopttheusualde nitionof(semi)stability,sothataGieseker-(orµ-)semistablesheafis

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

AFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACES5Proof.ForanycomplexstructureonXcompatiblewithitshyperk¨ahlerstructurethecurvatureoftheuniversalconnectionisoftype(1,1)[12].

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

6´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

thetwocomplexstructures,sothatonehasacommutativediagram

0 →

(2)

0 →∞FIDI ∞ S ) ∞ S+) →0 →π (F →π (FII g′ g′; II II ∞FI′ →π (F

hereDIisactuallytheadjointoftheDiracoperatorassociatedwiththecomplexstructureIcoupledwiththeconnectionofπ F Q;moreover,∞meansthatweareconsideringthesheafofsmoothsectionsofaholomorphicbundle.Theoperators ∞isthesheafofD aresurjectiveduetotheIT1condition,andeverysheafFI I(cf.[3,6]).smoothsectionsoftheFourier-MukaitransformsF

I→F I′ofC∞vectorbundles;Thediagram(2)inducesanisomorphismg II′:F

moreover,sincethebundleπ F Qhasanaturalhermitianmetric,thehorizontal Ioneveryarrowsinthisdiagramallowonetointroduceanhermitianmetrich I,andgbundleF II′isthenanisometry.Bycouplingtherelativeconnectioninducedbytheconnectiononπ F QwiththerelativeDolbeaultoperatorassociatedwith IacomplexstructureIandbytakingdirectimagesonede nesaconnection .SincethisconnectionisalsoinducedbythedirectimagesofthecoupledonF ∞ S±),diagram(2)provesthatg Iinto I′.connectionsonπ (F II′transforms II , Soactuallyonehasasinglehermitianbundle(Fh)withasingleconnection whosecurvatureisoftype(1,1)withrespecttoallcompatiblecomplexstructures. , )isaninstanton.Asaconsequence,theFourier-MukaiThen,thepair(F IofFIisµ-polystablewithrespecttotheK¨ deter-transformFahlerformoverXminedbyI. ∞ SI′) ∞ S+) →0 →π (FI DI′

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

AFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACES7Lemma1.IfXisare exiveK3surfaceandD·H>2foreverynodalcurveD,thedivisorE= +2Hisnote ective.Then,Hi(X,OX( +2H))=0fori≥0.Proof.SinceE2= 4,ifEise ective,itisnotirreducibleandE=D+FforsomenodalcurveD.ThenD·H=3sothatF·H=1andFisalsoirreducible.ItfollowsthatF2≥ 2.IfF2≥0,thenD·F≤ 1,sothatD=Fwhichisabsurd.Thus,F2= 2andFisanodalcurveofdegree1,asituationweareexcluding.

donotsatisfythelowerboundsontheOneshouldnoticethattheelementsinX

discriminantestablishedbySorger3andHirschowitzandLaszlo[11].

～X.Weconsidernoware exiveK3surfaceX →3.2.TheisomorphismX

satisfyingtheassumptiononnodalcurvesdescribedintheprevioussection,and =MH(v).showthatthereisanaturalisomorphismbetweenXandX

ThefollowingresultisadirectconsequenceofLemma1.

Lemma2.dimExt1(Ip( +2H),OX)=dimH1(X,Ip( +2H))=1foreverypointp∈X,whereIpistheidealsheafofp.

→X,givenTheseresultsimplythatthereexistsaone-to-onemapofsetsΨ:X

byΨ([E])=p,wherepisthepointdeterminedbyLemma3.Ournextaimistoprovethatthismapisactuallyanisomorphismofschemes;tothisendweneedaresultthatfollowsfromGrauert’sbasechangetheoremandLemma2.

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

8´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

Corollary1.ThesheafOX(H)isIT0,anditsFourier-MukaitransformN= .π (Q π OX(H))isalinebundleonXCorollary2.LetEbeasheafwhich tsintoanexactsequence

0 →OX →E(H) →Ip( +2H) →0,

whereIpistheidealsheafofapointp∈X.ThenEisµ-stableandlocallyfreewithv(E)=v=(2, , 3)andΨ([E])=p.

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

AFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACES9

= πwhereH (γ2,2H). isanaturalpolarizationonX .Indeed,theWewillshowthatthedivisorH ,regardedasamodulispaceofinstantonsonX,carriestheWeil-PeterssonspaceX

metricΦX ,alreadyconsideredinTheorem1,andwecanprove,inthespiritof

.[13],thattheclassofthismetricmaybeidenti edwiththeclassH =1Proposition6.H

2 i

X =0ΦX∧(c1(Q))2=2(4π)2( ·H)

ΦX∧trR2,0∧R0,2=0.

X

wemayByrepresentingtheCherncharacterγintermsofthecurvatureformR

compute

=1H

8π2 ΦX∧tr(2R2,0∧R0,2+R1,1∧R1,1)=1

X

Lemma4.Thedirectimageπ [Ext1(IΨ π OX( +2H),OX×X )]isalinebundle

.LonX

Proof.WriteE= +2HandOΨ=(ΓΨ) OX .Then,

Ext1(IΨ π OX(E),OX×X ) OΨ πOX( E). .Inthissectionweprovethatwe4.2.XasamodulispaceofbundlesonX 3) asamodulispaceofbundleswithtopologicalinvariants(2, ,canregardX ;itturnsoutthatthethatareµ-stablewithrespecttothenaturalpolarizationHrelevantuniversalbundleinthiscaseissimplyQ .Lemma3suggeststhattheuniversalsheafQcanbeobtainedasanextensionof .LetIΨbetheidealsheafofthesuitabletorsion-freerank-onesheavesonX×X →X×X ofΨ.graphΓΨ:X ·H isample,andcanbetakenasapolarizationonX ;moreover, =0,ThusH 2= 12,sothatv 3)isanisotropicMukaivectorandX isare exiveand =(2, , ). K3surfacewithrespectto(H,

ByLemma1,Riπ π OX( E)=0fori≥0,hence,fromtheexactsequence

0 →IΨ π OX( E) →π OX( E) →OΨ π OX( E) →0,

～R1π weobtainπ (OΨ π OX( E))→ (IΨ π OX( E)).Butforeveryξ∈X

onehasH1(X,IΨ π OX( E) κ(ξ))=H1(X,Ip( E)),wherep=Ψ(ξ),and

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

10´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

ItfollowsthatthesheafExt1(IΨ π OX( +2H),π (L 1))hasasection,sothatthereisanextension

(4)0 →π (L 1) →P →IΨ π OX( +2H) →0.

;Moreover,Lemma3impliesthatP π OX( H)isauniversalsheafonX×X～Q π .ThesheavesLandNthusP→ N π OX(H)foralinebundleNonX

arereadilydetermined;byapplyingπ tothesequenceaboveoneobtains

～ ( H ) N,L 1=OX(H) N→OX

wherethesecondequalityisduetoequation(3).Now,byrestrictingtheexact H c1(Q|π 1(p))= H .sequence(4)toa breπ 1(p),weobtainc1(N)=

Thenwehave

Proposition7.ThesequenceofcoherentsheavesonX×X

(5)

→Q π 0 →π OX OX→IΨ π OX( +2H) →0 ( 2H) ( H) πOX(H)

isexact.

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

AFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACES11

)impliesthatthemodulispaceM ( H, (X,Hv)ofstablesheavesonX(withrespect )withMukaivectorv ),toH isanon-emptyconnectedcomponentofSpl( v,Xconsistingoflocallyfreeµ-stablesheaves.

v)thesheafF tsintoanexactAccordingtotheproofofLemma3,if[F]∈MH (

unlessFisgivenbyanextensionsequencelike(7)forawell-de nedpointξ∈X

=1andZisazero-dimensionalclosedwhereDisanodalcurvewithD·H +2H .Inthelattercase,Hi(X, IZ( D))=0fori≥0sosubschemeofX +2H = 3andD·Ψ H= D)2.Then,D· thatlenght(Z)=0and 4=( )= 1,whichisabsurdsinceΨ Hisample.Then,onehas +2 D·(5H

～Q withp=Ψ(ξ),sincedimExt1(Iξ( +2H ,andF→ ),O )=forapointξ∈XpXv)iscontainedinα(X)andthetwospacesmustcoincide.This1.Thus,MH (

meansthatthebundlesQ pareµ-stablewithrespecttoH;thereforethesequence

withinvariants(5)exhibitsexplicitlytheparametrizationofvectorbundlesonX 3)thatareµ-stablewithrespecttoH bythepointsofX.Asaconsequence:(2, , +2H ) ) 0 →OX→F(H→Iξ( →0, +2H ) D) 0 →OX→F(H→IZ( →0 (D)

(polarizedbyProposition8.Xisa nemodulispaceofµ-stablebundlesonX 3),andtherelevantuniversalsheafisQ . )withinvariants(2, ,H

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

12´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

)thereisafunctorialisomorphismProposition10.ForeveryG∈D(X

).Moreover,ifXisre exiveandD·H>2foreveryinthederivedcategoryD(X

nodalcurveDinX,thenforeveryF∈D(X)thereisalsoafunctorialisomorphism

～SX (SX(F))→F[ 2]～SX(SX (G))→G[ 2]

inthederivedcategoryD(X).

×X ,andπijtheProof.Letq1andq2betheprojectionsontothetwofactorsofX

×X ontotheproductoftheithandjthfactors.Then,theprojectionofX×X

L )(see[18]),withcompositefunctorisgivenbySX(S (G))=Rq2, (q G QX1

whereRHom (,)denotesthetotalderivedfunctorofthecomplexofsheafho-momorphisms.By[20],Proposition4.10,theright-handsideisisomorphicinthe →X ×X isthediagonalembeddingderivedcategorytoδ M[ 2],whereδ:X

L .ItfollowsthatSX(S (G)) G M[ 2].LetandMisaninvertiblesheafonXX ρ:X×X→X×Xbethepermutationmorphism;bybase-changetheorywehave

byrelativedualityforπ23.Fromρ δ=ρone ndsthatM[ 2] M [ 2].Then,thereisanisomorphismofinvertiblesheavesM M ,andM OX .

Thesecondstatementfollowsfromthe rstbyProposition8. Rπ23, (π13 [ 2],ρQQ π12Q) Rπ23, [(π13Q π12Q ) ] Q LLL～Rπ23, RHom (π Q,π Q), =Rπ23, (π Q π Q)→Q12131213

,inacompleteanalogyThus,thereisadualitybetweenthevarietiesXandX

withthecaseoftheFourier-Mukaitransformonabeliansurfaces.

Inparticularwehavethefollowingresult.

Theorem2.LetFbeaWITisheafonX.ThenitsFourier-Mukaitransform =Riπ ,whoseFourier-MukaitransformF (π F Q)isaWIT2 isheafonX Q )isisomorphictoF.R2 iπ ( π F

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

AFOURIER-MUKAITRANSFORMFORSTABLEBUNDLESONK3SURFACES13

,F )forh=0,1,2.Inparticular,thereisanisomorphismExth(F,F) Exth(F issimpleforeverysimpleWITisheafF.foreveryh,sothatFLemma5.LetF,F′becoherentsheavesonX.IfFisWITiandF′isWITj,wehave ,F ′).Exth(F,F′) Exth+i j(F

isµ-polystablebyTheorem1,andsimplebyLemma5,sothatitisProof.F

µ-stable.Theorem3.LetFbeazero-degreeµ-stablebundleonX,withv(F )=(2, , 3). isµ-stable.ThenitsFourier-MukaitransformF

Corollary8.TheFourier-MukaitransformpreservestheEulercharacteristicand )=χ(F)andc1(F)·H=thedegreeofWIT1-sheaves,thatis,ifFisWIT1thenχ(F )·H .c1(F

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

14´´C.BARTOCCI,U.BRUZZOANDD.HERNANDEZRUIPEREZ

holomorphicsymplecticstructuresofthesespaces.Moreover,onehasdimMH(u)=µu)(sinceu2=u 2),andso—providedMH(u)isnotempty—thereisadimMH (

birationalcorrespondenceMH(u)→MHu). (

Insomecasesstrongerresultscanbeobtained;forinstanceitcanbeshownthat foranyn≥1themodulispaceMH (1+2n, n ,1 3n)isbiholomorphictothepunctualHilbertschemeHilbn(X)[8].

In[4]wegiveacompletelyalgebraicproofofTheorem3.Alsoin[4]weprove algebraicallythatOX (2H)isthedeterminantlinebundle,whichisanalternative

.proofoftheamplenessofH

ThetranscendentalproofofTheorem1extendsdirectlytohigherdimensions,providingaproofofthethefactthattheFourier-Mukaitransformonhyperk¨ahlermanifoldsmapsquaternionicinstantonstoquaternionicinstantons[6].

Acknowledgments.WethankP.Francia,J.M.Mu nozPorras,K.O’GradyandespeciallyA.Maciociaforusefuldiscussionsandsuggestions.The rstauthoralsothanksS.Donaldsonforadviceatapreliminarystageofthiswork.Thisworkwaspartlydonewhilethe rstauthorwasvisitingtheStateUniversityofNewYorkatStonyBrook,andthesecondauthorwasvisitingVictoriaUniversityofWellington,NewZealand.TheythanktherespectiveDepartmentsofMathematicsfortheirwarmhospitalityandforprovidingexcellentworkingconditions.

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We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli sp

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[17]15MamoneCapriaM.,SalamonS.M.,Yang-Mills eldsonquaternionicspaces,Nonlinearity1(1988),517–530.MaruyamaM.,ModuliofstablesheavesI,J.Math.KyotoUniv.17(1977),91–126.

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