Quot schemes and Ricci semipositivity C.R. Acad. Sci. Paris, Ser.I

Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Analytic geometry

Quot schemes and Ricci semipositivity

Schéma quot et semi-positivité de Ricci

Indranil Biswas

^a

, Harish Seshadri

^b

aSchoolofMathematics,TataInstituteofFundamentalResearch,HomiBhabhaRoad,Mumbai400005,India bIndianInstituteofScience,DepartmentofMathematics,Bangalore560003,India

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received19October2016

Acceptedafterrevision22March2017 Availableonline29March2017 PresentedbyJean-PierreDemailly

LetXbeacompactconnectedRiemannsurfaceofgenusatleasttwo,andletQ^X(r,d)be thequotschemethatparameterizesallthetorsioncoherentquotientsofO^⊕X^r ofdegreed.

ThisQX(r,d)isalsoamodulispaceofvorticesonX.Itsgeometricpropertieshavebeen extensivelystudied.Hereweprovethat theanticanonical linebundle ofQ^X(r,d) isnot nef. Equivalently, QX(r,d) does not admit any Kähler metric whose Ricci curvatureis semipositive.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

r é s um é

Soit X une surfacede Riemann compacte etconnexe de genre aumoins deux, etsoit QX(r,d) le schéma quot qui paramétrise tous les quotients torsion cohérents de O^⊕X^r de degré d.L’espace Q^X(r,d) est aussi un espace de modules de vortex sur X. Nous démontronsquelefibréanticanoniquedeXn’apaslapropriéténef.Defaçonéquivalente, QX(r,d)n’admetaucunemétriquekählériennedontlacourburedeRiccisoitsemi-positive.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

Takeacompact connectedRiemannsurface X.The genusof X,whichwillbe denotedby g,isassumedtobe atleast two.Wewillnotdistinguishbetweentheholomorphicvectorbundleson X andthetorsion-freecoherentanalyticsheaves on X.Forapositiveintegerr,letO^⊕X^r bethetrivialholomorphicvectorbundleon X ofrankr.Fixingapositiveintegerd, let

Q

:=

QX

(

r

,

d

)

(1.1)

bethequotschemethatparametrizesall(torsion)coherentquotientsofO^⊕X^r ofrankzeroanddegreed[17].Equivalently, QparametrizesallcoherentsubsheavesofO^⊕_X^{rofrankranddegree}−^{d,becausethesearepreciselythekernelsofcoherent}

E-mailaddresses:[email protected](I. Biswas),[email protected](H. Seshadri).

http://dx.doi.org/10.1016/j.crma.2017.03.012

1631-073X/©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

quotientsofO^⊕_X^rofrankzeroanddegreed.ThisQisaconnectedsmoothcomplexprojectivevarietyofdimensionrd.See [6,5,4] forthe propertiesof Q^{.It should be mentioned that} Q ^{isalso amoduli space ofvortices on X, andit hasbeen} extensivelystudiedfromthispointofviewofmathematicalphysics;see[3,9,12]andreferencestherein.

Bökstedt andRomão proved some interesting differential geometric properties of Q (see [12]). In [10] and [11],we proved thatQ^{doesnotadmitKählermetricswith}semipositiveorseminegativeholomorphicbisectionalcurvature.Inthis note, wecontinue tostudythequestionoftheexistence ofmetricsonQ ^{whosecurvaturehasasign.Ouraimhereisto} provethefollowing.

Theorem1.1.ThequotschemeQin(1.1)doesnotadmitanyKählermetricsuchthattheanticanonicallinebundleK_Q⁻¹isHermitian semipositive.

Since semipositive holomorphic bisectional curvature implies semipositive Ricci curvature for a Kähler metric, Theo- rem 1.1generalizesthemainresultof[11].

Recall that a holomorphic line bundle L on a compact complex manifold M is said to be Hermitiansemipositive if L admitsasmoothHermitianstructuresuchthatthecorrespondingHermitianconnectionhasthepropertythatitscurvature formissemipositive. Theanticanonicallinebundle onM willbe denotedby K_M⁻¹.Notethatif M admitsa Kählermetric such that thecorresponding Riccicurvatureissemipositive, then K⁻_M¹ is Hermitiansemipositive. Indeed,in thatcase, the Hermitianconnectionon K_M⁻¹fortheHermitianstructureinducedbysuchaKählermetrichassemipositivecurvature.The conversestatement,thattheHermitiansemipositivityofK_M⁻¹impliestheexistenceofKählermetricswithsemipositiveRicci curvature,isalsotruebyYau’ssolutiontotheCalabi’sconjecture[1,2,19].

TheproofofTheorem 1.1isbasedonarecentworkofDemailly,Campana,andPeternellontheclassificationofcompact Kähler manifolds M withsemipositiveK⁻_M¹ [15,14].Thisclassification impliesthat if K⁻_M¹ issemipositive,then thereis a nontrivialAbelianidealintheLiealgebraofholomorphicvectorfieldsonM,providedb1(M)>0.Ontheother hand,for M=Q^{,thisLiealgebraisisomorphicto}sl(r,C)^{,whichdoesnothaveanynontrivialAbelianideal.}

2. ProofofTheorem 1.1 2.1. SemipositiveRiccicurvature

Let J^d(X)=^Picd(X)betheconnectedcomponentofthePicardgroupof X thatparameterizestheisomorphismclasses of holomorphiclinebundles on X ofdegree d.Let S^d(X) denote thespaceof alleffective divisorson X of degreed, so S^d(X)= ^Xd/P_d isthesymmetricproduct,with P_d beingthegroupofpermutationsof{¹,· · ·,d}^.Let

p

:

^Sd

(

X

) −→

^Picd

(

X

)

(2.1)

bethenaturalmorphismthatsendsadivisoronX totheholomorphiclinebundleon X definedbyit.

Takeanycoherentsubsheaf F ⊂O^⊕_X^{rofrankranddegree}−^d.Let s_F

:

O^⊕_X^r

= (

O^⊕_X^r

)

^∗

−→

^F∗

bethedualoftheinclusionofF inO^⊕_X^r.Itsexteriorproduct

r

s_F

:

OX

=

r

O^⊕_X^r

−→

r

F^∗

isaholomorphicsectionoftheholomorphiclinebundle_r

F^∗ ofdegreed.Therefore,thedivisordiv(_r

s_F)isanelement of S^d(X).Consequently,wehaveamorphism

ϕ :

Q

−→

^Sd

(

X

) ,

F

−→

^div

(

r

s_F

) ,

(2.2)

whereQisdefinedin(1.1).Wenotethatwhenr=^1,then

ϕ

isanisomorphism.

AssumethatQ^{admitsaKählermetric}

ω

suchthat K_Q⁻¹isHermitiansemipositive.Thenthereisaconnectedfiniteétale Galoiscovering

f

:

_Q

−→

Q (2.3)

suchthat(Q, ^f∗

ω

)isholomorphicallyisometrictoaproduct

γ :

_Q

−→

A

×

C

×

H

×

F

,

(2.4)

where

• ^{AisanAbelianvariety,}

• ^{C isasimplyconnectedCalabi–Yaumanifold(holonomyisSU}(c),wherec=^dimC),

• ^{Hisasimplyconnected}hyper-Kählermanifold(holonomyisSp(h/2),whereh=^dimH),and

• ^{F isarationallyconnectedsmoothprojectivevarietysuchthat K−}_F^{1 isHermitian}semipositive.

(See [15,Theorem 3.1].) Henceforth,we willidentify Qwith A×^C×^H×^{F using}

γ

in(2.4). We note that F issimply connectedbecauseitisrationallyconnected[13,p. 545,Theorem3.5],[18,p. 362,Proposition2.3].

2.2. Alowerboundofd

Weknowthatb₁(Q)=^{2g,andtheinduced}homomorphism

(

p

◦ ϕ )

_∗

:

^H1

(

Q

, Q) −→

^H1

(

Pic^d

(

X

), Q) ,

where pand

ϕ

are constructedin(2.1)and(2.2),respectively,isanisomorphism[5],[6,p. 649,Remark].Since f in(2.3) isafiniteétalecovering,theinducedhomomorphism

f_∗

:

^H1

(

_Q

, Q) −→

^H1

(

Q

, Q)

issurjective.Therefore,thehomomorphism

(

p

◦ ϕ ◦

f

)

_∗

:

H1

(

_Q

, Q) −→

H1

(

Pic^d

(

X

), Q)

(2.5)

issurjective.

Thereisnononconstant holomorphicmapfromacompact simplyconnectedKählermanifoldtoan Abelianvariety.In particular,therearenononconstantholomorphicmapsfromC,HandFin(2.4)toPic^d(X).Hence,themapp◦

ϕ

◦^{f factors} throughamap

β :

^A

−→

^Picd

(

X

) .

Inotherwords,thereisacommutativediagram

Q

=

^A

×

^C

×

^H

×

^{F p}

−→

^{◦ϕ◦f Picd}

(

X

)

q

⏐⏐

^Id

A

−→

^{β Picd}

(

X

)

(2.6)

whereq istheprojectionofA×^C×^H×^{F tothefirstfactor.SinceH}1(A×^C×^H×^F,Z)= ^H1(A,Z)(asC,H andF are simplyconnected),and(p◦

ϕ

◦^f)_∗ in(2.5)issurjective,itfollowsthatthehomomorphism

β

_∗

:

^H1

(

A

, Q) −→

^H1

(

Pic^d

(

X

), Q)

inducedbyβ issurjective.Thisimmediatelyimpliesthatthemap β issurjective.Sinceβ issurjective,fromthecommuta- tivityof(2.6)weknowthatthemap pissurjective.Thisimpliesthat

d

=

^dimSd

(

X

) ≥

^{dim Picd}

(

X

) =

^g

≥

²

.

(2.7)

2.3. AlbaneseforQ

Thehomomorphismoffundamentalgroups

ϕ

_∗

: π

1

(

Q

) −→ π

1

(

S^d

(

X

))

induced by

ϕ

in(2.2) isanisomorphism[8, Proposition4.1].Since d≥^{2 (see(2.7)),the} homomorphismoffundamental groups

p_∗

: π

1

(

S^d

(

X

)) −→ π

1

(

Pic^d

(

X

))

inducedby pin(2.1)isanisomorphism.Indeed,π1(S^d(X))istheAbelianization

π

1

(

X

)/[π

1

(

X

), π

1

(

X

)] =

^H1

(

X

, Z)

ofπ1(X)[16].Combiningtheseweconcludethatthehomomorphismoffundamentalgroups

(

p

◦ ϕ )

_∗

: π

1

(

Q

) −→ π

1

(

Pic^d

(

X

))

(2.8)

inducedby p◦

ϕ

isanisomorphism.

Since the homomorphismin (2.8) is an isomorphism,the covering f in (2.3) is induced by a covering ofPic^d(X).In otherwords,thereisafiniteétaleGaloiscovering

μ _:

J

−→

^Picd

(

X

)

(2.9) andamorphismλ:Q−→ ^{J suchthatthefollowingdiagramis}commutative:

Q

−→

^f Q

⏐⏐ λ ⏐⏐

^p

◦ ϕ

J

−→

^{μ Picd}

(

X

)

(2.10)

where f isthecoveringmap in(2.3).Theprojectionqin(2.6)isclearly theAlbanesemorphismforQ^{,becauseC, Hand} F are all simplyconnected. Onthe other hand, p◦

ϕ

isthe Albanesemorphism for Q ^{[11, Corollary 2.2].Therefore, its} pullback,namely,λ,istheAlbanesemorphismforQ^.Consequently,wehave A = ^{J with}λcoincidingwiththeprojection q in(2.6).Henceforth,wewillidentify Aandq with Jandλrespectively.

2.4. Vectorfields

Thedifferentialdf of f identifiesTQwith f^∗TQ,because f isétale.Usingthetracehomomorphismt : ^f∗OQ −→O_Q, wehave

f_∗T_Q

=

^f∗f^∗TQ

−→

^pf

(

f_∗O_Q

) ⊗

^TQ

−→

^t O_Q

⊗

^TQ

=

^TQ

,

where p_f isgivenbytheprojectionformula.Thisproducesahomomorphism

:

H⁰

(

_Q

,

T_Q

) =

H⁰

(

Q

,

f_∗T_Q

) −→

H⁰

(

Q

,

TQ

)

(2.11)

(the equality H⁰(Q,TQ)= ^H0(Q, f_∗TQ) followsfrom thefact that f is afinite morphism). This homomorphism is surjective.Indeed,as f^∗TQ=^TQ,anysectionofTQpullsbacktoasectionofTQ.

SinceQ= ^A×^C×^H×^F,wehave

H⁰

(

_Q

,

T_Q

) =

^H0

(

A

,

T A

) ⊕

^H0

(

C

,

T C

) ⊕

^H0

(

H

,

T H

) ⊕

^H0

(

F

,

T F

) .

(2.12) Notethat H⁰(Q, TQ) isaLiealgebraundertheoperationofLiebracketofvectorfields,andthesubspace

H⁰

(

A

,

T A

) ⊂

H⁰

(

_Q

,

T_Q

)

(see(2.12))isanidealinthisLiealgebra.Since A = ^{JisacoveringofPicd}(X),wehave

dimH⁰

(

A

,

T A

) =

^{dim Picd}

(

X

) =

^g

>

1

.

(2.13)

Since H⁰(A,T A)isanidealinH⁰(Q, ^TQ)^,itfollowsimmediatelythat

(

H⁰

(

A

,

T A

)) ⊂ (

H⁰

(

_Q

,

T_Q

)) =

^H0

(

Q

,

TQ

)

isanideal,whereisconstructedin(2.11).NotethatH⁰(A,T A)isanAbelianLiealgebra,sotheLiealgebra(H⁰(A,T A)) isalsoAbelian.

Since

μ

: ^J = ^A −→^Picd(X) in (2.9) is a covering map between Abelian varieties, the trace map H⁰(A,T A)−→

H⁰(Pic^d(X),TPic^d(X))isanisomorphism.Inviewofthis,fromthecommutativityofthediagram in(2.10),itfollowsthat therestriction

|

H⁰(A,T A)

:

^H0

(

A

,

T A

) −→

^H0

(

Q

,

TQ

)

isinjective(see (2.12)and(2.11)). ButH⁰(Q,^TQ)=sl(r,C)^{[7,p. 1446, Theorem1.1].Hence theLiealgebra H0}(Q,^TQ) doesnotcontainanynonzeroAbelianideal.Thisisincontradictionwiththeearlierresultthat(H⁰(A,T A))isanonzero AbelianidealinH⁰(Q,TQ)ofdimension g(see(2.13)).ThiscompletestheproofofTheorem 1.1.

References

[1]T.Aubin,ÉquationsdutypeMonge–Ampèresurlesvariétéskählériennescompactes,C.R.Acad.Sci.Paris,Ser.I283(1976)119–121.

[2]T.Aubin,ÉquationsdutypeMonge–Ampèresurlesvariétéskählériennescompactes,Bull.Sci.Math.102(1978)63–95.

[3]J.M.Baptista,OntheL²-metricofvortexmodulispaces,Nucl.Phys.B844(2011)308–333.

[4]A.Bertram,G.Daskalopoulos,R.Wentworth,GromovinvariantsforholomorphicmapsfromRiemannsurfacestoGrassmannians,J.Amer.Math.Soc.9 (1996)529–571.

[5]E.Bifet,SurlespointsfixesschémaQuot_OX/X,ksousl’actiondutoreG^r_m,k,C.R.Acad.Sci.Paris,Ser.I309(1989)609–612.

[6]E.Bifet,F.Ghione,M.Letizia,OntheAbel–Jacobimapfordivisorsofhigherrankonacurve,Math.Ann.299(1994)641–672.

[7]I.Biswas,A.Dhillon,J.Hurtubise,AutomorphismsoftheQuotschemesassociatedtocompactRiemannsurfaces,Int.Math.Res.Not.2015(2015) 1445–1460.

[8]I.Biswas,A.Dhillon,J.Hurtubise,R.A.Wentworth,AgeneralizedQuotschemeandmeromorphicvortices,Adv.Theor.Math.Phys.19(2015)905–921.

[9]I.Biswas,N.M.Romão,ModuliofvorticesandGrassmannmanifolds,Commun.Math.Phys.320(2013)1–20.

[10]I.Biswas,H.Seshadri,OntheKählerstructuresoverQuotschemes,Ill.J.Math.57(2013)1019–1024.

[11]I.Biswas,H.Seshadri,OntheKählerstructuresoverQuotschemesII,Ill.J.Math.58(2014)689–695.

[12]M.Bökstedt,N.M.Romão,Onthecurvatureofvortexmodulispaces,Math.Z.277(2014)549–573.

[13]F.Campana,OntwistorspacesoftheclassC^{,J.Differ.Geom.33(1991)541–549.}

[14]F.Campana,J.-P.Demailly,T.Peternell,RationallyconnectedmanifoldsandsemipositivityoftheRiccicurvature,in:C.D.Hacon,M.Musta ¸t˘a,M.Popa (Eds.),RecentAdvancesinAlgebraicGeometry,AVolumeinHonorofRobLazarsfeld’s60thBirthday,PapersfromtheConferenceHeldattheUniversity ofMichigan,AnnArbor,MI,May16–19,2013,in:Lond.Math.Soc.Lect.NoteSer.,vol. 417,CambridgeUniversityPress,Cambridge,UK,2015,pp. 71–91.

[15]J.-P.Demailly,StructuretheoremsforcompactKählermanifoldswithnefanticanonicalbundles,in:ComplexAnalysisandGeometry, in:Springer ProceedingsinMathematics&Statistics,vol. 144,Springer,Tokyo,2015,pp. 119–133.

[16]A.Dold,R.Thom,QuasifaserungenundunendlichesymmetrischeProdukte,Ann.ofMath.(2)67(1958)239–281.

[17]A.Grothendieck,Techniquesdeconstructionetthéorèmesd’existenceengéométriealgébrique,IV.LesschémasdeHilbert,SéminaireBourbaki,vol. 6, SociétémathématiquedeFrance,Paris,1995,pp. 249–276,Exp.No.221.

[18]J.Kollár,Fundamentalgroupsofrationallyconnectedvarieties,Mich.Math.J.48(2000)359–368.

[19]S.-T.Yau,OntheRiccicurvatureofacomplexKählermanifoldandthecomplexMonge–AmpèreequationI,Commun.PureAppl.Math.31(1978) 339–411.