A symplectic analog of the Quot scheme C.R. Acad. Sci. Paris, Ser.I

Contents lists available atScienceDirect

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

www.sciencedirect.com

Algebraic geometry

A symplectic analog of the Quot scheme

Un analogue symplectique du schéma Quot

Indranil Biswas

^a

, Ajneet Dhillon

^b

, Jacques Hurtubise

^c

, Richard A. Wentworth

^d

aSchoolofMathematics,TataInstituteofFundamentalResearch,HomiBhabhaRoad,Bombay400005,India bDepartmentofMathematics,MiddlesexCollege,UniversityofWesternOntario,London,ONN6A5B7,Canada cDepartmentofMathematics,McGillUniversity,BurnsideHall,805SherbrookeSt.W.,Montreal,QC H3A0B9,Canada dDepartmentofMathematics,UniversityofMaryland,CollegePark,MD20742,USA

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

Articlehistory:

Received16June2015

Acceptedafterrevision11September2015 Availableonline20October2015 PresentedbyClaireVoisin Keywords:

SymplecticQuotscheme Automorphism Symmetricproduct

We construct a symplectic analog of the Quot scheme that parameterizes the torsion quotientsofatrivialvectorbundleoveracompactRiemannsurface.Someofitsproperties areinvestigated.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

r é s um é

NousconstruisonsunanaloguesymplectiqueduschémaQuot,quiparamètrelesmodules quotientsdetorsiond’unfibrévectorieltrivialsurunesurfacedeRiemanncompacte,et nousexaminonscertainesdesespropriétés.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

1. Introduction

Let X beacompactconnectedRiemannsurface.Forfixedintegersnandd,letQ

(

O^⊕_Xⁿ

,

d

)

denotetheQuotschemethat parameterizesthetorsionquotientsofO^⊕_X^{nofdegreed(see[8]for}constructionandpropertiesofageneralQuotschemes).

ThisparticularQuotschemeQ

(

O^⊕Xⁿ

,

d

)

arisesinthestudyofmodulispaceofvectorbundlesofranknon X [4,7,3].Being amodulispaceofvortices,itisalsostudiedinmathematicalphysics(see[1]andreferencestherein).

Here weconsidera symplectic analogofthe Quotscheme.LetO^⊕_X^{2r bethetrivialvector bundleon X equippedwith} a symplectic structure given by the standard symplectic form on C^{2r.Take torsion quotients ofit of degree dr that are} compatiblewiththe symplecticstructure(thisisexplainedinSection 2.1).Fixing X,r andd,let Q^{denotetheassociated} symplecticQuotscheme.TheprojectivesymplecticgroupPSp

(

2r

,

C

)

hasanaturalactiononQ.Weshowthattheconnected component,containingtheidentityelement,ofthegroupofallautomorphismsofQ^{coincideswithPSp}

(

2r

,

C)^.

In[4],Bifet,GhioneandLetiziausedtheusualQuotschemesQ(O^⊕_Xⁿ

,

d

)

associatedwithX tocomputethecohomologies ofthemodulispacesofsemistablevectorbundlesofrankn on X.Ourhopeistobeabletocompute thecohomologies of themodulispaceofsemistableSp

(

2r

,

C

)

-bundleson X usingthesymplecticQuotschemeQ^.

E-mailaddresses:[email protected](I. Biswas),[email protected](A. Dhillon),[email protected](J. Hurtubise),[email protected] (R.A. Wentworth).

http://dx.doi.org/10.1016/j.crma.2015.09.006

1631-073X/©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

2. ThesymplecticQuotscheme

2.1. ConstructionofthesymplecticQuotscheme Let

ω

:=

r

i=¹

(

e^∗_i

⊗

^e∗_i+r

−

^e∗_i+r

⊗

^e∗_i

)

bethestandardsymplecticformonC^{2r.LetXbeacompactconnectedRiemannsurface.Thesheafof}holomorphicfunctions on X willbedenotedbyOX.Let

E0

:=

O^⊕_X^2r

be thetrivialholomorphicvectorbundleon X ofrank2r.Theabovesymplecticform

ω

definesasymplecticstructureon E0,becausethefibersof E0 areidentifiedwithC^{2r.Thissymplecticstructureon E}0 willbedenotedby

ω

0.

Fixanintegerd≥^1.Let

Q

:=

Q

( ω

0

,

d

)

(1)

bethesymplecticQuotschemeparameterizingalltorsionquotients

τ

Q

:

E0

−→

Q

ofdegreedr satisfyingthefollowingcondition:there isaneffectivedivisorDQ on X ofdegreedsuch thattherestricted form

ω

0|kernel(τQ)factorsas

kernel

( τ

Q

) ⊗

kernel

( τ

Q

) −→

^ω0 OX

(−

DQ

) →

OX

;

(2)

it should be clarified that DQ depends on Q. Equivalently, the symplectic Quot scheme Q parameterizes all coherent analyticsubsheaves F⊂^E0ofrank2randdegree−^{drsuchthattheform}

ω

0|F factorsas

F

⊗

^F

−→

^ω0 OX

( −

^D

) →

OX

,

where D issomeeffectivedivisoron X ofdegreedthatdependsonF.ThisdescriptionofQ^{sendsanysubsheaf F tothe} quotientsheaf E0

/

F.

Theabovepairing

F

⊗

^F

−→

^ω0 OX

( −

^D

)

producesaninjectivehomomorphism

μ :

^F

−→

OX

(−

^D

) ⊗

^{F∗ (3)}

betweencoherentanalyticsheavesofrank2r;thehomomorphism

μ

isinjectivebecauseitisinjectiveoverthecomplement ofthesupportofE0

/

F.Since

degree

(

F

) = −

^dr

=

^degree

(

OX

( −

^D

) ⊗

^F∗

) ,

the homomorphism

μ

in (3) is an isomorphism. This means that the pairing Fx⊗^Fx ω0(x)

−→OX

(

−^D

)

x is nondegenerate for every x∈^{X. The divisor D is uniquely determined by F because}

μ

isan isomorphism.More precisely, consider the homomorphismofcoherentanalyticsheaves

μ :

^F

−→

^F∗

givenbytherestriction

ω

0|F.ThedivisorD istheschemetheoreticsupportofthequotientsheaf F^∗

/ μ (

F

)

. Thegroupofallpermutationsof{¹

,

· · ·

,

d}^{willbedenotedby S}d.Thequotient

X^d

/

S_d (4)

of X^d forthenaturalactionofS_disthesymmetricproductSym^d

(

X

)

.Let

ϕ :

Q

−→

^Symd

(

X

)

(5)

bethemorphismthatsendsanyquotientQ tothedivisorDQ (see(2)).

Let

Q

:=

Q

(

O^⊕_X^2r

,

rd

)

(6)

betheQuotschemethatparameterizesalltorsionquotientsofO^⊕_X^2r=^E0 ofdegreerd.Itisanirreduciblesmoothcomplex projectivevarietyofdimension2r²d.Foranysubsheaf F ofE0 withE0

/

F∈Q^,wehave

T_E0/FQ

=

^H0

(

X

, (

E₀

/

F

) ⊗

^F∗

) .

(7)

Nowassumethat E0

/

F∈Q^{.Firstconsiderthe}homomorphismF⊗^E0 ω0

−→OX obtainedbyrestricting

ω

0.Notethat

ω

0

(

F⊗ F

)

⊂OX

(

−

ϕ (

E0

/

F

))

,where

ϕ

isthemorphismin(5)(see(2)).Therefore,

ω

0 producesahomomorphism

θ :

F

⊗ (

E0

/

F

) −→

OX

/(

OX

(− ϕ (

E0

/

F

))) .

Thesubspace

T_E0/FQ

⊂

^TE0/F_Q

consistsofallhomomorphisms

α

:^F−→^E0

/

F (see(7))suchthat

θ (

v

⊗ α (

w

)) = θ (

w

⊗ α (

v

)) .

ClearlyQisaclosedsubschemeofQ.

Lemma1.TheschemeQisanirreducibleprojectivevarietyofdimensiond

(

r²+^r+²

)/

2.

Proof. The morphism

ϕ

in (5)is clearly surjective. Also, each fiberof

ϕ

isirreducible. Since Sym^d

(

X

)

is irreducible,we concludethatQ^isalsoirreducible.

Takeapointx:= {^x1· · ·

,

xd}∈^Symd

(

X

)

suchthatallxi∈^X,1≤ⁱ≤^{d,aredistinct.Let}

L ⊂

^Gr

( C

^2r

,

r

)

(8)

bethevarietyparameterizingallLagrangiansubspacesofC^{2rforthestandardsymplecticform}

ω

.WehaveamapL^d−→

ϕ

⁻¹

(

x

)

thatsendsany

(

V1

,

· · ·

,

Vd

)

∈L^dtothecomposition O^2r_X

−→

O^2r_X

|

x1∪···∪xd

=

d

i=¹

C

^2rx_i

−→

d

i=¹

C

^2rx_i

/

V_i

,

where C^2rxi is the sheaf supported at x_i withstalk C^{2r; note that this} composition is surjective and henceit defines an element of

ϕ

⁻¹

(

^x

)

.Thismap L^d−→

ϕ

⁻¹

(

^x

)

isclearly an isomorphism.SincedimL=^r

(

r+¹

)/

2, itfollowsthat dimQ= d

(

r²+^r+²

)/

2. 2

2.2. Vortexequationandstability

FixaHermitianmetric

ω

X on X;notethat

ω

X isKähler.

Takeasubsheaf F⊂O^⊕_X^{2r suchthatthequotient}O^⊕_X^2r

/

F isatorsionsheafofdegreedr,so F∈Q^(see(6)).Let

O^⊕_X^2r

→

^{F∗ (9)}

bethedualoftheinclusionofF inO^⊕_X^{2r.Notethatthe}homomorphismin(9)definesa2r-pairofrank2r[2,p.535(3.1)].

Therefore,eachelementofQ^{definesa2r-pairofrank2r.}

Takeanyrealnumber

τ

,andtake anelement zF∈Q.Let F⊂O^⊕_X^2r be thesubsheaf representedby zF.Wenote that thesubsheaf F is

τ

-stableifrd

< τ

(see[2,p.535,Definition3.3]forthedefinitionof

τ

-stability).

Weassume that

τ >

rd.Therefore,allelements ofQ are

τ

-stable. Henceevery F⊂O^⊕_X^2r lying inQ admitsaunique Hermitianstructurethatsatisfiesthe2r–

τ

-vortexequation[2,p.536,Theorem3.5].

TakeanyF⊂O^⊕X^2rrepresentedbyanelementofQ^.LethF denotetheuniqueHermitianstructureonF thatsatisfiesthe 2r–

τ

-vortexequation.Theisomorphism

μ

in(3)takesh_F totheuniqueHermitianstructureonOX

(−

^D

)

⊗^{F∗ thatsatisfies} the 2r–

τ

-vortex equation forOX

(

−^D

)

⊗^F∗∈Q^{.Indeed,thisfollows} immediatelyfromtheuniqueness of theHermitian structuresatisfyingthe2r–

τ

-vortexequation.

3. AutomorphismsofthesymplecticQuotscheme

Inthissectionweassumethatgenus

(

X

)

≥^2.

ThegroupofallholomorphicautomorphismsofQwillbedenotedbyAut

(Q)

.Let

Aut

(

Q

)

⁰

⊂

^Aut

(

Q

)

be theconnectedcomponentof itcontaining theidentity element.The group oflinearautomorphismsofthe symplectic vector space

(

C^2r

, ω

)

is Sp

(

2r

,

C

)

. The standard action of Sp

(

2r

,

C

)

on C^{2r produces an action ofSp}

(

2r

,

C

)

on Q. The centerZ/²Z^ofSp

(

2r

,

C)^{actstriviallyon}Q^{.Therefore,wegeta}homomorphism

ρ :

^PSp

(

2r

, C) =

^Sp

(

2r

,C)/(Z/

²

Z) −→

^Aut

(

Q

)

⁰

.

Theorem2.Theabovehomomorphism

ρ

isanisomorphism.

Proof. ThestandardactionofSp

(

2r

,

C

)

onC^{2rproducesanactionofPSp}

(

2r

,

C

)

onthevarietyL^{in(8).Thisactionon}L^is effective.Fromthisitfollowsimmediatelythatthehomomorphism

ρ

isinjective(recallthatthegeneralfiberof

ϕ

isL^d).

Foreach 1≤ⁱ≤^{d,let p}i:^Xd−→^{X betheprojection tothei-thfactoroftheCartesianproduct.Foreachpair1}≤ⁱ

<

j≤^d,let

i,j

⊂

^Xd

bethedivisoroverwhichthetwomapsp_i andp_jcoincide.Let

U

:=

^Xd

\ (

1≤ⁱ<j≤^d i,j

)

be thecomplement.Consider allpoint{^x1· · ·

,

xd}∈^Symd

(

X

)

such thatxk=^x forallk=

.ThecomplementinSym^d

(

X

)

ofthe subsetdefinedby suchpoints willbe denotedby U.Thequotientmap X^d−→^Xd

/

S_d (see(4)) sends U to U.The quotientmap

f

:

U

:=

^Xd

\ (

1≤i<j≤d

i,j

) −→

^{U (10)}

isanétaleGaloiscoveringwithGaloisgroup S_d.

Theinverseimage

ϕ

⁻¹

(

U

)

⊂Q^{willbedenotedby}V^,where

ϕ

istheprojectionin(5).Let

ϕ

:= ϕ |

_V

:

V

−→

U (11)

be therestrictionof

ϕ

.WenotethatV ^{isafiberbundleoverU withfibersisomorphicto}L^{d (see(8)for}L^).Inparticular, V ^{iscontainedinthesmoothlocusof}Q^.

The Lie algebra of Sp

(

2r

,

C

)

will be denoted by sp

(

2r

,

C

)

. The Lie algebra of Aut

(

Q

)

⁰ is contained in the space of algebraicvectorfieldsH⁰

(V,

TV)equippedwiththeLiebracketoperationofvectorfields.Let

d

ρ :

^sp

(

2r

,C) −→

^H0

(

V

,

TV

)

(12)

bethehomomorphismofLiealgebrasassociatedwiththehomomorphism

ρ

ofLiegroups. Toprovethat

ρ

issurjective,it sufficestoshowthatd

ρ

issurjective.Wenotethatd

ρ

isinjectivebecause

ρ

isinjective.

Takeanyalgebraicvectorfield

γ ∈

^H0

(

V

,

TV

) .

Let

d

ϕ

_:

TV

−→ ϕ

^∗T U

bethedifferentialoftheprojection

ϕ

in(11).Asnotedbefore,thefibersof

ϕ

areisomorphictoL^d,inparticular,theyare connectedsmoothprojectivevarieties,soanyholomorphicfunctiononafiberof

ϕ

isaconstantfunction.Thisimpliesthat thesectiond

ϕ

( γ )

descendstoU.Inotherwords,thereisaholomorphicvectorfield

γ

onU suchthat

d

ϕ

( γ ) = ϕ

^∗

γ

.

(13)

Let

γ

:=

^f∗

γ

∈

^H0

(

U

,

T

U

)

(14)

be the pullback, where f is the projection in (10). Since the vector field

γ

is algebraic, the above vector field

γ

is meromorphicon X^d,meaning

γ

∈

^H0

(

X^d

, (

T X^d

) ⊗

O_X^d

(

1≤ⁱ<j≤^d

m

·

i,j

))

forsomeintegerm.Itisknownthattherearenosuchnonzerosections[5,Proposition2.3].Therefore,wehave

γ

₌0,and hencefrom(13)and(14)itfollowsthat

d

ϕ

( γ ) =

⁰

.

(15)

ThestandardactionofSp

(

2r

,

C)^onC^{2rproducesanactionofSp}

(

2r

,

C)^onL^{definedin(8).Let} sp

(

2r

, C ) −→

^H0

( L ,

T

L )

bethecorrespondinghomomorphismofLiealgebras.Itisknownthattheabovehomomorphismisanisomorphism.Let

T_rel

−→

U

×

UV

−→

U

betherelativealgebraictangentbundlefortheprojectionU×UV−→U.Since

U

×

UV

=

_U

× L

^d

,

andH⁰

(

U

,

OU

)

=C^{[5,Lemma2.2],wehave}

H⁰

(

U

×

UV

,

T_rel

) =

^H0

(L,

^T

L)

^⊕d

=

^sp

(

2r

,C)

^⊕d

.

(16) Let

TV

⊃

^Tϕ

−→

V

betherelative algebraictangentbundle fortheprojection

ϕ

.Since f in(10)isaGaloisétalecovering withGaloisgroup Sd,from(16)wehave

H⁰

(

V

,

T_ϕ

) =

^H0

(

U

×

UV

,

T_rel

)

^Sd

= (

sp

(

2r

, C )

^⊕d

)

^Sd

=

^sp

(

2r

, C ) .

Combiningthiswith(15)weconcludethat

H⁰

(

V

,

TV

) =

^sp

(

2r

, C) .

Inparticularthehomomorphismd

ρ

in(12)issurjective. 2

Corollary3.TheisomorphismclassofthevarietyQ^{uniquelydeterminesthe}isomorphismclassofX ,exceptwhend=²=^genus

(

X

)

. Proof. ThisfollowsfromTheorem 2and[6].TheargumentissimilartotheproofofTheorem5.1in[5]. 2

References

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

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

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

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

[5]I.Biswas,A.Dhillon,J.Hurtubise,Automorphisms oftheQuot schemesassociated tocompactRiemannsurfaces,Int.Math.Res. Not.2015(2015) 1445–1460.

[6]N.Fakhruddin,Torelli’stheoremforhighdegreesymmetricproductsofcurves,arXiv:math/0208180v1.

[7]F.Ghione,M.Letizia,EffectivedivisorsofhigherrankonacurveandtheSiegelformula,Compos.Math.83(1992)147–159.

[8]A.Grothendieck,Techniquesdeconstructionetthéorèmesd’existenceengéométriealgébrique.IV.LesschémasdeHilbert,in:SéminaireBourbaki, vol. 6,1960–1961,pp. 249–276,exp.n^o221.