K-stability of constant scalar curvature Kahler manifolds

manifolds

arXiv:0803.4095v1 [math.AG] 28 Mar 2008JacopoStoppa

Abstract

WeshowthatapolarisedmanifoldwithaconstantscalarcurvatureK¨ahlermetricanddiscreteautomorphismsisK-stable.ThisrefinestheK-semistabilityprovedbyS.K.Donaldson.

1Introduction

Let(X,L)beapolarisedmanifold.OneofthemorestrikingrealisationsinK¨ahlergeometryoverthepastfewyearsisthatifonecanfindaconstantscalarcurvatureK¨ahler(cscK)metricgonXwhose(1,1)-formωgbelongstothecohomologyclassc1(L)then(X,L)issemistable,inanumberofsenses.TheseminalreferencesareYau[16],Tian[13],Donaldson[5],[7].

InthisnoteweareconcernedwithDonaldson’salgebraicK-stability[7],seealsoDefinition2.5below.ThisnotiongeneralisesTian’sK-stabilityforFanomanifolds[13].ItshouldplayarolesimilartoMumford-Takemotoslopestabilityforbundles.ThenecessarygeneraltheoryisrecalledinSection2.AsymptoticChowstability(whichimpliesK-semistability,seee.g.[12]Theorem3.9)foracscKpolarisedmanifoldwasfirstprovedbyDonaldson[6]intheabsenceofcontinousautomorphisms.Importantworkinthiscon-nectionwasalsodonebyMabuchi,seee.g.[10].FromtheanalyticpointofviewthefundamentalresultisthelowerboundontheK-energyprovedbyChen-Tian[4].

TheneatestresultinthealgebraiccontextseemstobeDonaldson’slowerboundontheCalabifunctional,whichwenowrecall.

󰀁itsaverageForaK¨ahlerformωletS(ω)denotethescalarcurvature,S

(atopologicalquantity).DenotebyFtheDonaldson-Futakiinvariantofatestconfiguration(Definitions2.1,2.2).Theprecisedefinitionofthenorm󰀞X󰀞appearingbelowwillnotbeimportantforus.

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Theorem1.1(Donaldson[8])Forapolarisedmanifold(X,L)

󰀃

󰀁)2ωn≥−supXF(X)(S(ω)−Sinf

ω∈c1(L)

X

cscKpropertyisopen-atleastinthesenseofsmalldeformations-socscKshouldimplystability.Ofcourseweneedtomakethisrigorous;inparticulartestingsmalldeformationsisnotenoughtoproveK-stability.

Thussupposethat(X,L)isproperlyK-semistable(Definition2.7).WewillfindanaturalwaytoconstructfromthisafamilyofK-unstablesmallperturbations(Xε,Lε)forsmallε>0.OurchoiceforXεisactuallyconstant,

󰀁=BlqXataveryspecialpointqwithexceptionaldivisorE.theblowupX

Onlythepolarisationchanges,andquitenaturallyLε=π∗L−εO(E).Thiswouldinvolvetakingε∈Q+andworkingwithQ-divisors,butinfactwe

󰀁polarisedbyLγ=π∗Lγ−O(E)rathertaketensorpowersandworkwithX

forintegerγ≫0.K-(semi,poly,in)stabilityisunaffectedbyDefinition2.2.Proposition1.5Let(X,L)beaproperlyK-semistablepolarisedmanifold.Thenthereexistsapointq∈Xsuchthatthepolarisedblowup(BlqX,π∗Lγ⊗O(−E))isK-unstableforγ≫0.

Remark1.6ItisinterestingtonotethatthecorrespondingresultforvectorbundlesfollowsfromBuchdahl[3].Let(X,L)beapolarisedmanifoldandE→Xaproperlyslopesemistablevectorbundle.Thenthepullbackπ∗EtotheblowupBlq1,...qmXinafinitenumberofsuitablychosenpointsisslopeunstablewithrespecttothepolarisationπ∗Lγ⊗OBl{qi}X(1)forγ≫0.Assumenowthataproperlysemistable(X,L)alsoadmitsacscKmetricω∈c1(L).IfAut(X,L)isdiscretetheblowupperturbationproblemforωisunobstructedbyatheoremofArezzoandPacard[1],sowewouldgetcscKmetricsinc1(π∗Lγ⊗O(−E))forγ≫0,acontradiction.

Remark1.7Thisperturbationstrategyforproving1.3isverygeneral,andwasfirstpointedouttotheauthorbyS.DonaldsonandG.Sz´ekelyhidi.Differentchoicesfor(Xε,Lε)leadtodifferentperturbationproblemsforω,whichmaysettleConjecture1.3inthepresenceofcontinousautomorphisms.ApossiblevariantistoperturbthecscKequationwithεatthesametime,butonewouldthenneedtodeveloptherelevantK-stabilitytheoryforamoregeneralequation.

Tosumupthemainingredientsforourproof(besidesTheorem1.1)are:1.Awellknownembeddingresultfortestconfigurations(Proposition2.9),togetherwiththealgebro-geometricestimateProposition3.3;

3

2.AblowupformulafortheDonaldson-Futakiinvariantprovedbytheauthor[14]Theorem1.3;3.AspecialcaseoftheresultsofArezzoandPacardonblowingupcscKmetrics[1].Aknowledgements.IamgratefultoS.K.Donaldson,G.Sz´ekelyhidiandmyadvisorR.Thomasformanyusefuldiscussions.Thereference[3]waspointedouttomebyJ.Keller.

2Somegeneraltheory

LetndenotethecomplexdimensionofX.

Definition2.1(Testconfiguration.)Atestconfigurationforapolarisedmanifold(X,L)isapolarisedflatfamily(X,L)→Cwith(X1,L1)∼=(X,L)

∗

andwhichisC-equivariantwithrespecttothenaturalactionofC∗onC.Givenatestconfiguration(X,L)for(X,L)denotebyAkthematrixrepre-sentationoftheinducedC∗-actiononH0(X0,Lk0).By(equivariant)Riemann-Rochwecanfindexpansions

nn−1

h0(X0,Lk+O(kn−2),0)=a0k+a1k

tr(Ak)=b0kn+1+b1kn+O(kn−1).

(2.1)(2.2)

Definition2.2(Donaldson-Futakiinvariant.)Thisistherationalnum-ber

2

F(X)=a−(2.3)0(b0a1−a0b1)whichisindependentofthechoiceofaliftingoftheactiontoL0.

EquivalentlyF(X)isthecoefficientofk−1intheLaurentseriesexpansionofthequotient

tr(Ak)

Remark2.3(Coverings)Givenatestconfiguration(X,L)wecancon-structanewtestconfigurationfor(X,L)bypullingXandLbackunderthed-foldramifiedcoveringofCgivenbyz→zd.ThischangesAktod·AkandconsequentlyFtod·F.

Definition2.4Atestconfiguration(X,L)iscalledaproductifitisC∗-equivariantlyisomorphictotheproduct(X×C,p∗XL)endowedwiththecompositionofaC∗-actionon(X,L)withthenaturalactionofC∗onC.Aproducttestconfigurationiscalledtrivialiftheassociatedactionon(X,L)istrivial.

TheDonaldson-FutakiinvariantF(X)inthiscasecoincideswiththeclassicalFutakiinvariantforholomorphicvectorfields.

Definition2.5(K-stability)Apolarisedmanifold(X,L)isK-semistableifforalltestconfigurations(X,L)

F(X)≥0.

ItisK-stableifthestrictinequalityholdsfornontrivialtestconfigurations.Inparticularif(X,L)isK-stableAut(X,L)mustbediscrete.ThecorrectnotiontotakecareofcontinousautomorphismsisK-polystability.

Definition2.6Apolarisedmanifold(X,L)isK-polystableifitisK-semista-bleandmoreoveranytestconfiguration(X,L)withF(X)=0isaproduct.Definition2.7Apolarisedmanifold(X,L)isproperlyK-semistableifitisK-semistableanditadmitsanonproducttestconfigurationwithvanishingDonaldson-Futakiinvariant.

Remark2.8TheterminologystrictlyK-semistableisalsofoundinthelit-eraturewiththesamemeaning.

Testconfigurationsarewellknowntobeequivalentto1-parameterflatfam-iliesinducedbyprojectiveembeddings.

Proposition2.9(seee.g.Ross-Thomas[12]3.7)Atestconfigurationfor(X,L)isequivalenttoa1-parametersubgroupofGL(H0(X,L)∗).In[14]theauthorprovedablowupformulafortheDonaldson-Futakiinvari-ant.Thestatementinvolvessomemoreterminology.

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Definition2.10(Hilbert-Mumfordweight.)Letαbea1-parametersub-groupofSL(N+1),inducingaC∗-actiononPN.Chooseprojectivecoordi-nates[x0:...:xN]suchthatαisgivenbyDiag(λm0,...λmN).TheHilbert-Mumfordweightofaclosedpointq∈PNisdefinedby

µ(q,α)=−min{mi:qi=0}.

NotethatthiscoincideswiththeweightoftheinducedactiononthefibreofthehyperplanelinebundleO(1)overthespecialisationlimλ→0λ·q.Definition2.11(Chowweight.)Let(Y,L)beapolarisedscheme,y∈Yaclosedpoint,andαaC∗-actionon(Y,L).SupposethatLisveryampleandα֒→SL(H0(Y,L)∗).TheChowweightCH(Y,L)(q,α)isdefinedtobetheHilbert-Mumfordweightofy∈P(H0(Y,L)∗)withrespecttotheinducedaction.Thedefinitionextendsto0-dimensionalcyclesonY,thatiseffectivelinearcombinationsofclosedpoints.

Theorem2.12(S.[14]1.3)Forpointsqi∈Xandintegersai>0letZ⊂Xbethe0-dimensional󰀄n−1closedsubschemeZ=∪iaiqi.LetΛbethe0-cycleonXgivenbyiaiqi.

A1-parametersubgroupα֒→Aut(X,L)inducesatestconfiguration󰀁,L󰀁)for(BlZX,π∗Lγ⊗OBlX(1)),whereOBlX(1)denotestheexceptional(XZZ

−

invertiblesheaf.MorepreciselyletO(Z)betheclosureoftheorbitofZ.

󰀁=π∗Lγ⊗Ob(1).󰀁=BlO(Z)−XandLThenXX

0

SupposethatαactsthroughSL(H(X,L)∗)withFutakiinvariantF(X).Thenthefollowingexpansionholdsasγ→∞

󰀁)=F(X)−CH(X,L)(Λ,α)F(X

γ1−n

2(n−1)!

6

+O(γ−n)

holdsasγ→∞.

WeemphasisethattherelevantChowweightiscomputedonthecentralfibre(X0,L0)withitsinducedC∗-action.

Proof.Theargumentof[14]Section4goesoververbatimtonon-producttestconfigurations,withonlytwoexceptions:

󰀁→X→C;1.TheproofofflatnessofthecompositionX2.TheidentificationoftheweightCH(X0,L0)(Λ0)(withrespecttothein-ducedactiononX0)withCH(X,L)(Λ,α).Wedonotneedthelatteridentification,andindeeditdoesnotmakesense

inthiscasesincethegeneralfibreisnotpreservedbytheC∗-action.

Toproveflatnessweusethecriterion[9]IIIProposition9.7.Thuswe

󰀁(i.e.irreduciblecomponentsneedtoprovethatallassociatedpointsofX

andtheirthickenings)maptothegenericpointofSpec(C).

ByflatnessthisistrueforthemorphismX→C,andblowingupO(Λ)−doesnotcontributenewassociatedpoints,onlytheCartierexceptionaldi-visorπ−1O(Λ)−.

−󰀁Moreprecisely󰀂letIddenotetheidealsheafofO(q)⊂X,andrecallXis

definedasProjd≥0I.Anyhomogeneouszerodivisorinthegradedsheaf󰀂d

d≥0IisalreadyazerodivisorwhenregardedasanelementofOX.On

󰀁isbydefinition(following[9]IIItheotherhandanassociatedpointx󰀁∈X

Corollary9.6)apointforwhicheveryelementofmxbisazerodivisor.The

󰀁→XmapsmxnaturalmapXbtoitsdegree0piece.Thusbytheabove

remarkx󰀁necessarilymapstoanassociatedpointx∈X.ButxmapstothegenericpointofSpec(C)byflatness,sothesameistrueforx󰀁.

Q.E.D.

Remark2.14InbothcasestheassumptionthatαactsthroughSLisnotreallyrestrictive.ThiscanalwaysbeachievedbyreplacingLbysomepowerandpullingbackXbyz→zdforsomed.ThisgivesanewtestconfigurationforwhichαcanberescaledtoactthroughSLandforwhichtheFutakiinvariantisonlymultipliedbyd,byRemark2.3.

ThispropertyoftheFutakiinvariantturnsouttobeimportantinourproofofTheorem1.2.

3ProofofTheorem1.2

ItwillbeenoughtoproveProposition1.5andtoapplytheresultofArezzoandPacardrecalledasTheorem3.1below.

7

󰀁Lγ)=(BlqX,π∗Lγ⊗O(−E)).(X,

󰀁Lγ)isK-unstableforγ≫0.WewillconstructtestWeneedtoshowthat(X,

󰀁Lγ)whichhavestrictlynegativeDonaldson-configurations(Xγ,Lγ)for(X,Futakiinvariantforγ≫0.

Byassumption(X,L)isproperlysemistable,soitadmitsanontrivialtestconfiguration(X,L)withF(X)=0.

MoreoverwecanassumethattheinducedC∗-actiononH0(X0,L0)∗isspeciallinear.IndeedthiscanbeachievedbytakingsomepowerLrandaramifiedcoverz→zd.ThenewFutakiinvariantF′stillvanishessinceF′=d·F=0.

WeblowXupalongtheclosureO(q)−oftheorbitO(q)ofq∈X1undertheC∗-actiononX,i.e.define

󰀁󰀁LetOXb(1)denotetheexceptionalinvertiblesheafonX.WeendowXwith

thepolarisation

Lγ=π∗Lγ⊗OX(3.2)b(1).Definetheclosedpointq0∈X0tobethespecialisation

q0=limλ·q.

λ→0

Thuslet

󰀁=BlO(q)−X.Xγ=X

(3.1)

Applyingtheblowupformula2.13inthiscasegives󰀁0,π∗LγF(X(1))=F(X0,L0)−CH(X0,L0)(q0)c0⊗OX0

2(n−2)!

CH(X0,L0)(q0)>0.

ThisholdsthankstotheassumptionF(X)=0,ormoregenerallyF(X)≤0.

ThisisenoughtosettleProposition1.5.

ThefinalstepforTheorem1.2istoshowthattheperturbationproblemisunobstructedprovidedAut(X,L)isdiscrete.ThisispreciselythecontentofabeautifulresultofC.ArezzoandF.Pacard.

8

γ1−n

+O(γ−n).

InProposition3.3belowwewillprovethatforaveryspecialq∈X1∼=X,

Theorem3.1(Arezzo-Pacard[1])Let(X,L)beapolarisedmanifoldwithacscKmetricintheclassc1(L).SupposeAut(X,L)isdiscreteandletq∈Xbeanypoint.ThentheblowupBlqXwithexceptionaldivisorEadmitsacscKmetricintheclassγπ∗c1(L)−c1(O(E))forγ≫0.

Remark3.2TheArezzo-PacardtheoremalsoholdsintheK¨ahlercaseand,moreimportantly,evenwhenaut(X,L)=0,providedasuitablestabilityconditionissatisfied.Wereferto[2],[14]forfurtherdiscussion.

ThusthefollowingPropositionwillcompleteourproof(s).Webelieveitmayalsobeofsomeindependentinterest.

Proposition3.3Let(X,L)beanonproducttestconfigurationforapo-larisedmanifold(X,L)withnonpositiveDonaldson-Futakiinvariantandsup-posetheinducedC∗-actiononH0(X0,L0)∗isspeciallinear.Thenthereexistsq∈X1∼=XsuchthatCH(X0,L0)(q0)>0.Proof.BytheembeddingTheorem2.9wereducetothecaseofanontrivialC∗actingonPNforsomeN,oftheformDiag(λm0x0,...λmNxN),orderedby

m0≤m1...≤mN.

Let{Zi}ki=1bethedistinctprojectiveweightspaces,whereZihasweightmi(i.e.theinducedactiononZiistrivialwithweightmi).EachZiisaprojectivesubspaceofPN,andthecentralfibrewithitsreducedin-red

ducedstructureX0isacontainedinSpan(Zi1,...Zil)forsomeminimalflag0=i1Thecase1λ→0

limλ·q=q0∈Zil.

SuchapointexistsbyminimalityandbecausethespecialisationofeverypointmustlieinsomeZj.SincetheactiononX0isinducedfromthatonPN,q0belongstothetotallyrepulsivefixedlocusR=X0∩Zil⊂X0.BythiswemeanthateveryclosedpointinX0\\RspecialisestoaclosedpointinX0\\R.In󰀂dC∗particularthenaturalbirationalmorphismX0󰀁󰀁󰀂Proj(dH0(X0,L⊗0))blowsupalongR.Soq0∈RisanunstablepointfortheC∗-actioninthesenseofgeometricinvarianttheory.BytheHilbert-MumfordcriteriontheweightoftheinducedactiononthelineL0|q0mustbestrictlypositive.SinceweareassumingthattheinducedactiononH0(X0,L0)∗isspeciallinearthis

9

weightcoincideswiththeChowweight,soCH(X0,L0)(q0)>0.

Degeneratecase.Intherestoftheproofwewillshowthatinthedegener-red

atecaseX0⊂Z0theDonaldson-Futakiinvariantisstrictlypositive.NotethatsincebyassumptiontheoriginalC∗-actiononPNisnontrivial,Z0⊂PNisaproperprojectivesubspace.

Wedigressforamomenttomakethefollowingobservation:foranyC-actiononPNwithorderedweights{mi},andasmoothnondegeneratemanifoldY⊂PN,themapρ:Y∋y→y0=limλ→0λ·yisrational,definedontheopendenseset{y∈Y:µ(y)=m0}ofpointswithminimalHilbert-Mumfordweight.Indeed,intheabovenotation,genericpointsspecialisetosomepointinthelowestfixedlocusZ0.InanycasethemapρblowsupexactlyalonglociwheretheHilbert-Mumfordweightjumps.

∗

red

GoingbacktoourdiscussionofthecaseX0⊂Z0,weseethatthismeanspreciselythatallpointsofX1haveminimalHilbert-Mumfordweightm0,sothereisawelldefinedmorphism

ρ:X1→Z0.

Moreoverρisafinitemap:thepullbackofL0underρisLwhichisam-ple,thereforeρcannotcontractapositivedimensionalsubscheme.Ifρwereanisomorphismonitsimage,itwouldfitinaC∗-equivariantisomorphismX∼=X×C.Thereforeρcannotbeinjective,eitheronclosedpointsortangentvectors.If,say,ρidentifiesdistinctpointsx1,x2,thismeansthatthexispecialisetothesamexundertheC∗-action;byflatnessthenthelocalringOX0,xcontainsanontrivialnilpotentpointingoutwardsofZ0,i.e.thesheafIX0∩Z0/IX0isnonzero.InotherwordsX0isnotaclosedsubschemeofZ0.Thecasewhenρannihilatesatangentvectorproducesthesamekindofnilpotentinthelocalringofthelimit,byspecialisation.

Tosumup,thecentralfibreX0isnonreduced,containingnontrivialZ0-orthogonalnilpotents.Equallyimportant,theinducedactionontheclosedsubschemeX0∩Z0⊂X0istrivial.Theproofwillbecompletedbyaweightcomputation.

Donaldson-Futakiinvariant.SupposeZ0⊂PNhasprojectivecoordinates[x1:...:xr],i.e.itiscutoutby{xr+1=...=xN=0}.WechangethelinearisationbychangingtherepresentationoftheC∗-action,tomakeitof

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theform

[x0:...xr:xr+1:...:xN]→[x0:...xr:λmr+1−m0xr+1:...:λmN−m0xN],

(3.3)

andrecallmr+i>m0foralli>0.ItispossiblethattheinducedactiononH0(X0,L0)∗willnotbespeciallinearanymore,howeverthisdoesnotaffecttheDonaldson-Futakiinvariant.Notethatforalllargek,

H0(PN,O(k))→H0(X0,Lk0)→H1

(IX0(k))=0.

(3.4)

By3.4,ourgeometricdescriptionofX0andthechoiceoflinearisation3.3

weseethatanysectionξ∈H0(X0,Lksectionξcan0)hasnonnegativeweightunderthe

inducedC∗

-action.Theonlyhavestrictlypositiveweightifitisoftheformxr+i·fforsomei>0.Moreoverweknowthereexistsanintegera>0suchthatxar+i|X0=0foralli>0.Letw(k)denotethetotalweightoftheactiononH0(X0,LkΛP(k)H0(X0),i.e.theinducedweightontheline

0,Lk0k

discussionimplies0),whereP(k)=h(X0,Ltheupperbound

0)istheHilbertpolynomial.Ourw(k)≤C(P(k−1)+...+P(k−a))

(3.5)

forsomeC>0,independentofk.Inparticular,

w(k)=O(kn).

(3.6)Ontheotherhandwecanlookatjustonesectionxr+i,i>0withxr+i|X0=0.Thisgivesalowerbound

w(k)≥C·P(k−1)

(3.7)

forsomeC>0,independentofk.Soweseethat

w(k)

k.

(3.8)holdsfork≫0andsomeC′>0independentofk.Togetherwith

w(k)

C′′

kP(k)

=

forsomeC′′>0independentofk.

BydefinitionofDonaldson-Futakiinvariant,thisimmediatelyimplies

F(X)≥C′′>0,

acontradiction.

Q.E.D.

Remark3.4Onecancharacterisethedegeneratecaseintheaboveproofmoreprecisely.

AsobservedbyRoss-Thomas[12]Section3aresultofMumfordimpliesthatanytestconfiguration(X,L)for(X,L)isacontractionofsomeblowupofX×CinaflagofC∗-invariantclosedsubschemessupportedinsomethickeningofX×{0}.

Theexistenceofthemapρ:X1→Z0meanspreciselythatinthisMumfordrepresentationofXnoblowupoccurs,i.e.XisacontractionoftheproductX×C.

Defineamapν:X×C→Xbyν(x,λ)=λ·xawayfromX×{0},ν=ρonX×{0}.Thisisawelldefinedmorphism,andsinceρisfinite,νispreciselythenormalisationofX.

red

SointhedegeneratecaseX0⊂Z0thenormalisationofXisX×C.Ross-Thomas[12]Proposition5.1provedthegeneralresultthatnormalis-ingatestconfigurationreducestheDonaldson-Futakiinvariant.ThisalreadyimpliesF≥0inthedegeneratecase,sincetheinducedactiononX×CmusthavevanishingFutakiinvariant.Inourspecialcaseourdirectproofyieldsthestrictinequalityweneed.

Remark3.5TheresultofMumfordmentionedabovestatesmorepreciselythatanytestconfiguration(X,L)for(X,L)isacontractionoftheblowupofX×Cinanidealsheaf

Ir=I0+tI1+...+tr−1Ir−1+(tr)

whereI0⊆...⊆Ir−1⊂OXcorrespondtoadescendingflagofclosedsubschemesZ0⊇...⊇Zr−1.Theactionon(X,L)isthenaturaloneinducedfromthetrivialactiononX×C.

SupposenowthatF(X)=0andthatnocontractionoccursinMumford’srepresentation.

TheninProposition3.3wecansimplychooseanyclosedpointq∈Zr−1.ThisisbecausethepropertransformofZr−1×CcutsX0inthetotallyrepulsivelocusfortheinducedaction,i.e.theactionflowseveryclosedpointinX0outsidethislocustothepropertransformofX×{0}.

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Converselyblowingupq∈X\\Z0onlyincreasestheDonaldson-Futakiinvariant(atleastasymptotically).

ForexampleK-stabilitywithrespecttotestconfigurationswithr=1andnocontractionisknownasRoss-Thomasslopestability[12]andhasfoundinterestingapplicationstocscKmetrics.InparticularthisdiscussiongivesasimplerproofthatacscKpolarisedmanifoldwithdiscreteautomorphismsisslopestable.

Remark3.6ArefinementofConjecture1.3wasproposedbyG.Sz´ekelyhidi.Ifω∈c1(L)iscscKthereshouldbeastrictlypositivelowerboundforasuit-ablenormalisationofFoverallnonproducttestconfigurations.Thiscondi-tioniscalleduniformK-polystability.In[15]Section3.1.1itisshownthatthecorrectnormalisationinthecaseofalgebraicsurfacescoincideswiththatofTheorem1.1,namelyF(X)

[4]Chen,X.,Tian,G.GeometryofK¨ahlermetricsandfoliationsbyholo-morphicdiscsarXiv:math/0507148v1[math.DG][5]Donaldson,S.K.Remarksongaugetheory,complexgeometryand4-manifoldtopology,FieldsMedallists’Lectures(M.F.AtiyahandD.Iagol-nitzer,eds.),WorldSci.Publ.,Singapore,1997,pp.384-403.[6]Donaldson,S.K.Scalarcurvatureandprojectiveembeddings.I.J.Dif-ferentialGeom.59(2001),no.3,479–522.[7]Donaldson,S.K.Scalarcurvatureandstabilityoftoricvarieties.J.Dif-ferentialGeom.62(2002),no.2,2–349.[8]Donaldson,S.K.LowerboundsontheCalabifunctional.J.DifferentialGeom.70(2005),no.3,453–472.[9]Hartshorne,R.AlgebraicGeometry.Springer-Verlag,NewYork,(1977).[10]Mabuchi,T.StabilityofextremalKhlermanifolds.OsakaJ.Math.41(2004),no.3,563–582.[11]Paul,S.andTian,G.CMStabilityAndTheGeneralisedFutakiInvari-antIIarXiv:math/0606505v2[math.DG][12]Ross,J.andThomas,R.P.AstudyoftheHilbert-Mumfordcriterionforthestabilityofprojectivevarieties.J.AlgebraicGeom.16(2007)201-255.arXiv:math/0412519v2(2004)[13]Tian,G.K¨ahler-Einsteinmetricswithpositivescalarcurvature.Invent.Math.137(1997),no.1,1–37.[14]Stoppa,J.Unstableblowups,J.AlgebraicGeom.(toappear)arXiv:math/07021v2[math.AG][15]Sz´ekelyhidi,G.ExtremalmetricsandK-stabilityPh.D.Thesis,ImperialCollegearXiv:math/0611002v1[math.DG][16]OpenproblemssectioninSchoen,R.,Yau,S.-T.Lecturesondifferentialgeometry.ConferenceProceedingsandLectureNotesinGeometryandTopology,I.InternationalPress,Cambridge,MA,1994.Universit`adiPavia,ViaFerrata127100Pavia,ITALYand

ImperialCollege,LondonSW72AZ,UK.E-mail:jacopo.stoppa@unipv.it

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