On the Chow ring of certain rational cohomology tori

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Algebraic geometry

On the Chow ring of certain rational cohomology tori

Sur l’anneau de Chow de certaines variétés dont la cohomologie rationnelle est celle d’un tore

Zhi Jiang

^a

^,

^b

, Qizheng Yin

^c

aDépartementdemathématiques,CNRSUMR8628,UniversitéParis-Sud,Bâtiment 425,91405Orsaycedex,France

bShanghaiCenterforMathematicalSciences,FudanUniversity,No. 220HandanRoad,YangpuDistrict,Shanghai200433,China cBeijingInternationalCenterforMathematicalResearch,PekingUniversity,No. 5YiheyuanRoad,HaidianDistrictBeijing100871,China

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

Articlehistory:

Received22March2017 Accepted18April2017 Availableonline28April2017 PresentedbyClaireVoisin

Let f : ^X→^{A be an abelian cover from a complex algebraic variety with quotient} singularities to an abelian variety. We show that f^∗ induces an isomorphism between the rational cohomology rings H^•(A,Q) and H^•(X,Q) if and only if f^∗ induces an isomorphismbetweentheChowringswithrationalcoefficientsCH^•(A)_QandCH^•(X)_Q.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Soit f:^X→^{Aun revêtementabélien d’unevariété algébrique complexeà}singularités- quotientdansunevariétéabélienne.Nousmontronsque f^∗induitunisomorphismeentre lesanneauxdecohomologierationnelleH^•(A,Q)etH^•(X,Q)sietseulementsi f^∗induit unisomorphismeentrelesanneauxdeChowàcoefficientsrationnelsCH^•(A)_QetCH^•(X)_Q.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. RationalcohomologytoriandtheChowring

Acomplexalgebraicvariety X iscalledarationalcohomologytorusifX isnormaland H^•

(

X

, Q ) ∧

^•H1

(

X

, Q ).

In[3],the authorsstudiedpropertiesof rationalcohomologytori. Theyshowedthatif X isa rationalcohomologytorus, then thereexists afinitecovera_X

:

^X

→

^{A toan abelianvariety suchthat a∗}_X

:

^H•

(

A

,

Q)

→

^H•

(

X

,

Q) ^isan isomorphism.

ThemorphismaX istheuniversalmorphismtoanabelianvarietyandiscalledtheAlbanesemorphismof X.

ItisageneralprinciplethatHodgestructurescontrolChowgroups.Henceitmakessensetoaskthefollowingquestion.

E-mailaddresses:[email protected],[email protected](Z. Jiang),[email protected](Q. Yin).

http://dx.doi.org/10.1016/j.crma.2017.04.006

1631-073X/©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Question1.1.Let f

:

^X

→

^{A be afinitemorphism froma varietywithquotient}singularitiestoan abelianvariety. Assume that f^∗

:

^H•

(

A

,

Q)

→

^H•

(

X

,

Q)^isanisomorphism.Does f^∗induceanisomorphismofChowringswithrationalcoefficients between Aand X?

A variety issaid tohave quotientsingularitiesifit isétale-locally the quotientofa smooth variety bya finite group.

For such varieties there is a well-defined intersection theory with Q-coefficients (see [7]). Note that a smooth rational cohomology toruscannot be ofgeneraltype,butinevery dimension

>

2, thereexistrationalcohomologytoriofgeneral typewithquotientsingularities(see[3]).HenceitismoreinterestingtoconsidertheChowringofrationalcohomologytori withquotientsingularities.

Despite severalconstraints onaX found in [3], we do not havea thorough classification ofrational cohomologytori.

Hence it seems difficult to answer the above question in general. But all known rational cohomology tori are actually abeliancoversofabelianvarieties(see[3,Section4]).Thegoalofthisnoteistoansweraffirmativelytheabovequestionin thecaseofabeliancoversofabelianvarieties.

Let f

:

^X

→

^{A bean abeliancoverfromanormalvarietytoanabelianvarietywithGaloisgroup G.Then wehave(see} forinstance[6])

f_∗OX

=

χ∈^G∗ L⁻_χ¹

,

where Lχ isalinebundleon A andG actson L⁻χ¹ viathecharacter

χ

.Assumefurther that X hasquotientsingularities.

Defineforeachcharacter

χ ∈

^{G∗theprojector}

[

_χ^X

] :=

¹

|

^G

|

g∈^G

χ

⁻¹

(

g

)[

g^X

] ∈

^CHdimX

(

X

×

^X

)

_C

,

where

_g^X isthegraphofg.TheimageofCH^•

(

X

)

_C under

[

_χ^X

]

^is CH^•

(

X

)

^χ_C

:= { α ∈

CH^•

(

X

)

_C

:

g_∗

( α ) = χ (

g

) α

for all g

∈

G

}.

Wehaveadecomposition CH^•

(

X

)

_C

=

χ∈G^∗

CH^•

(

X

)

^χ_C

.

Theorem1.2.Letf

:

^X

→

^{A beanabeliancoverfromavarietywithquotient}singularitiestoanabelianvarietywithGaloisgroupG.

Thenthefollowingstatementsareequivalent:

(1) X isarationalcohomologytorus;

(2)hⁱ

(

A

,

Lχ

) =

^0foralli

∈

Z^andnon-trivial

χ _∈

G^∗;

(3)

[

_χ^X

] =

⁰

∈

^CHdimX

(

X

×

^X

)

_Cforallnon-trivial

χ _∈

G^∗;inparticular, f^∗

:

^CH•

(

A

)

_Q

→

^CH•

(

X

)

_Q

isanisomorphism.

The implications (1)

⇒

^{(2) and(3)}

⇒

^(1)are straightforward. Indeed,if X is arational cohomologytorus, then f^∗

:

H^•

(

A

,

Q)

→

^H•

(

X

,

Q)^isanisomorphismofQ^-Hodgestructures.Wehaveforalli

∈

Z^,

hⁱ

(

A

,

OA

) =

^h0,i

(

A

) =

^h0,i

(

X

) =

^hi

(

X

,

OX

) =

χ∈^G∗

hⁱ

(

A

,

L⁻_χ¹

).

Hencehⁱ

(

A

,

L⁻χ¹

) =

^{0 foralli}

∈

Z^andnon-trivial

χ ∈

^{G∗.Wealsohavehi}

(

A

,

Lχ

) =

^{0 bySerreduality.}

If

[

_χ^X

] =

⁰

∈

^CHdimX

(

X

×

^X

)

_C for all non-trivial

χ _∈

G^∗, then by applying

[

_χ^X

]

^{to H•}

(

X

,

C

)

, we find H^•

(

X

,

C

) =

H^•

(

X

,

C

)

^G.Hence f^∗

:

^H•

(

A

,

Q

) →

^H•

(

X

,

Q

)

isanisomorphismandX isarationalcohomologytorus.

Theproofof(2)

⇒

^{(3)willoccupythenextsection.}

Remark1.3.Since Lχ is semi-ample (see Step 1 in the proof of Proposition 2.2), condition (2) in Theorem 1.2 is also equivalent totheconditionthath⁰

(

A

,

Lχ

) =

^{0 forall}non-trivial

χ ∈

^{G∗.Hence,undertheassumptionof}Theorem 1.2,we knowthat X isarationalcohomologytorusifandonlyifpg

(

X

) =

^1.

2. ProofofTheorem 1.2

Lemma2.1.LetA beanabelianvarietyandlett

:

^A

→

^{A bethe}translationbyatorsionelement.Then

[

_t^A

] −[

A

] =

⁰

∈

^CHdimA

(

A

×

A

)

_Q,where

_t^Aisthegraphoft and

Aisthediagonal.

Proof. Letm

∈

Z>0 be theorder of t. Recallthat m^∗

:

^CH•

(

A

×

^A

)

_Q

→

^CH•

(

A

×

^A

)

_Q is an isomorphism(see [1]). Since m^∗

[

_t^A

] =

^m∗

[

A

] ∈

^CHdimA

(

A

×

^A

)

_Q,weconcludethat

[

_t^A

] − [

A

] =

⁰

∈

^CHdimA

(

A

×

^A

)

_Q. 2

Proposition2.2.Let

ρ :

^Y

→

^{A beacycliccoverfromavarietywithquotient}singularitiestoanabelianvarietywithGaloisgroupH . Wewrite

ρ

_∗OY

=

χ∈H^∗

L⁻_χ¹

.

Ifhⁱ

(

A

,

Lχ

) =

^0foralli

∈

Z^andnon-trivial

χ _∈

H^∗,thenforallnon-trivial

χ _∈

H^∗,

h∈H

χ

⁻¹

(

h

)[

_h^Y

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_C

.

Proof. We proceed by induction on the dimension of Y. The statement in dimension 0 is trivial. We then assume that Proposition 2.2holdsindimensions

<

dimY.

Step1.Reductiontolowerdimensions.

Let

χ

be ageneratorof H^∗.Sincehⁱ

(

A

,

Lχ

) =

^{0 foralli}

∈

Z^{,thelinebundle L}χ isnotample.LetK denotetheneutral componentofthemorphism

ϕ

L_χ

:

^A

→

^Pic0

(

A

)

inducedbyLχ,andlet p

:

^A

→

^{B bethequotientby K.Thenwehave}

L_χ

=

p^∗H

+

Q

,

whereHisanamplelinebundleonBandQ isatorsionlinebundlesuchthat Q

|

K

=

OK (seeforinstance[2,Section 3.3]).

As

ρ

isacycliccoverand

χ

isageneratorofH^∗,foreachk

∈

Z^{,thereexistsaneffectivedivisorD}kwhosecomponentsare irreduciblecomponentsofthebrancheddivisorDsuchthat

kL_χ

=

^L_χ^k

+

^Dk

.

Moreover,form

= |

^H

|

^{,weknowthatmL}χ

=

^DmandthatDm

−

^{Diseffective(see[6]).In}particular,wehaveD

=

^p∗DB for someampledivisorDB onB.

ConsidertheSteinfactorizationofthenaturalmorphism

ϕ =

^p

◦ ρ :

^Y

→

^B,

ϕ :

^Y

−→

^{ϕ Z}

−−→

^{ϕ B}

.

Ageneralfiberof

ϕ

isanétalecoverofK,whichisanabelianvarietythatwedenotebyK.Hence

ϕ

istheIitakafibration ofY inthesenseof[5,Theorem13].

LetY

:=

^Z

×

BA.Thenwehaveafactorization

ρ :

Y ^ρ

−→

Y ^ρ

−−→

A

.

Both

ρ

and

ρ

arecycliccoverswithrespectiveGaloisgroupsH

⊂

^HandH

:=

^H

/

H.Hencethemorphism

ϕ

:

^Z

→

^Bis alsoan H-cover.Moreover,wewrite

ϕ

_∗OZ

=

ι∈^{H ∗} L⁻_ι,¹_Z

.

Then we have p^∗Lι,Z

=

^Lχι,where

χ

_ι is the induced character of H. Hence Hⁱ

(

B

,

Lι,Z

) =

^{0 for all i}

∈

Z ^and non-trivial

ι ∈

^{H ∗.}

Further,weshowthatY andZ havequotientsingularities.Recallfrom[7,Proposition 2.8]thatY isthecoarsemoduli space ofa canonical smooth Deligne–Mumford stack Y^{. By the universal property of the canonical stack (see [4, Theo-} rem 4.6]),theactionof HonY inducesanactiononY^{.ThequotientY}

=

^Y

/

Histhecoarsemodulispaceofthesmooth Deligne–Mumfordstack

[Y /

H

]

^{,andhencehasquotient}singularities.

We take a quotient A

→

^{K such that A}

→

^B

×

^{K isan isogeny.Each connectedcomponent ofa generalfiber ofthe} inducedmorphismY

→

^{K isanétalecoverofZ.Hence Z alsohasquotient}singularities.

SincedimZ

<

dimY,theinductionhypothesisimpliesthatforallnon-trivial

ι ∈

^{H ∗,}

h∈^H

ι

⁻¹

(

h

) [

_h^Z

] =

⁰

∈

^CHdimZ

(

Z

×

^Z

)

_C

.

Step2.Proofthat

h∈H

ι

⁻¹

(

h

) [

^Y_h

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_Cforallnon-trivial

ι _∈

H ^∗.

Consider the quotient A

→

^{K such that A}

→

^B

×

^{K is an isogeny withGalois group L. Then the induced morphism} π

:

^Y

=

^Z

×

BA

→

^Z

×

^{K isalsoanétalecoverwithGaloisgroupL.}

Notethatforallh

∈

^H,

( π × π )

^∗

[

_h^Z

×

K

] =

l∈^L

[

_h^Z

×

B

_l^A

] ∈

^CHdimY

(

Y

×

^Y

)

_Q

.

Consider foreachl

∈

^{Lthe embeddings}

_h^Z

×

B

_l^A

⊂

_h^Z

×

B

(

A

×

B A

) ⊂

^Y

×

^{Y.ByLemma 2.1,we have}

[

_l^A

] = [

A

] ∈

CH_dimY

(

A

×

B A

)

_Q.Hence

[

_h^Z

×

B

_l^A

] = [

_h^Z

×

B

A

] ∈

^CHdimY

(

_h^Z

×

B

(

A

×

BA

))

_Q

.

Inparticular,

[

_h^Z

×

B

_l^A

] = [

_h^Z

×

B

_A

] ∈

^CHdimY

(

Y

×

^Y

)

_Q

.

Thenwefind

(π × π)

^∗

h∈H

ι

⁻¹

(

h

)[

_h^Z

×

K

]

=

h∈H

ι

⁻¹

(

h

)

l∈L

[

_h^Z

×

B

_l^A

]

= |

^L

|

h∈^H

ι

⁻¹

(

h

) [

_h^Z

×

B

_A

]

= |

L

|

h∈H

ι

⁻¹

(

h

)[

_h^Y

] ∈

CH^dimY

(

Y

×

Y

)

_C

.

As

h∈H

ι

⁻¹

(

h

) [

_h^Z

×

K

] =

⁰

∈

^CHdimY

(

Z

×

^Z

×

^K

×

^K

)

_C,wealsohave

h∈^H

ι

⁻¹

(

h

) [

_h^Y

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_C

.

Step3.Proofthat

h∈^H

χ

⁻¹

(

h

) [

_h^Y

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_Cforallnon-trivial

χ ∈

^{H ∗.}

ByKawamata’stheorem[5,Theorem13],thereexistsanétalecover

μ :

Y

→

^{Y withGaloisgroup J inducedbyanétale} coverofA,suchthatthefollowingholds:considertheSteinfactorizationofthenaturalmorphism

ν = ϕ

_◦ μ :

Y

→

^Z,

ν :

Y

−→

^ν

Z

−→

^{ν Z}

.

ThenwehaveY

Z

×

K with

ν

theprojection, JactsonZfaithfullyandonK faithfullybytranslation,andY

(

Z

×

K

)/

J.

Weputeverythingintoacommutativediagram:

Y ν

μ Y

ρ ρ ϕ

Y ^ρ A

p

Z ^ν Z ^ϕ B

.

Since

μ

isinduced by an étalecoverof A,we knowthat

ρ

_◦ μ :

Y

→

^{Y isstill an abeliancover. Itis clearthat the} Galoisgroupof

ρ ◦ μ

is J

×

^{HandhencetheGaloisgroupof}

ρ

◦ μ

is J

×

^H.Letq

:

^J

×

^H

→

^Hbetheprojection.

Considerthemorphism

μ × μ :

Y

×

Y

→

^Y

×

^{Y.Notethatforallh}

∈

^H,

( μ _× μ )

^∗

[

_h^Y

] =

g∈^q−1(h)

[

_g^Y

] ∈

^CHdimY

(

Y

×

Y

)

_Q

.

Henceforallnon-trivial

χ ∈

^{H ∗,}

( μ _× μ )

^∗

(

h∈^H

χ

⁻¹

(

h

) [

_h^Y

] ) =

(g,h)∈^J×^H

χ

⁻¹

(

h

) [

₍^Y_g,h)

] ∈

^CHdimY

(

Y

×

Y

)

_C

.

UndertheisomorphismY

Z

×

K,wehave

[

^Y_(g,h)

] = [

^Z_g

×

₍^K_g,h)

] ∈

^CHdimY

(

Z

×

Z

×

K

×

K

)

_Q

.

Thenwefind

( μ _× μ )

^∗

(

h∈^H

χ

⁻¹

(

h

) [

_h^Y

] ) =

(g,h)∈^J×^H

χ

⁻¹

(

h

) [

_g^Z

×

₍^K_g,h)

]

=

g∈^J

([

_g^Z

] × (

h∈H

χ

⁻¹

(

h

)[

₍^K_g,h)

])) ∈

CH^dimY

(

Z

×

Z

×

K

×

K

)

_C

.

ByLemma 2.1,wehave

[

₍^K_g,h)

] = [

_K

] ∈

^CHdimK

(

K

×

K

)

_Q.Hence

h∈^H

χ

⁻¹

(

h

) [

₍^K_g,h)

] =

⁰

∈

^CHdimK

(

K

×

K

)

_C

,

andthus

( μ _× μ )

^∗

(

h∈H

χ

⁻¹

(

h

) [

_h^Y

] ) =

⁰

∈

^CHdimY

(

Y

×

Y

)

_C.Bytheprojectionformula,

h∈H

χ

⁻¹

(

h

)[

_h^Y

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_C

.

Step4.Conclusion.

Let

χ _∈

H^∗ be a non-trivial character. If therestriction of

χ

to H is trivial, then

χ

is the pull-backof a non-trivial character

ι

ofH.Asbefore,

h∈H

χ

⁻¹

(

h

)[

_h^Y

] = ( ρ

× ρ

)

^∗

(

h∈H

ι

⁻¹

(

h

)[

_h^Y

]) ∈

^CHdimY

(

Y

×

^Y

)

_C

.

ByStep2,weseethat

h∈^H

χ

⁻¹

(

h

) [

_h^Y

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_C.

Ifthe restrictionof

χ

to H is non-trivial,we take a subset V of H consistingofrepresentatives ofeach coset of H. Thenwehave

h∈H

χ

⁻¹

(

h

)[

_h^Y

] = (

h∈H

χ

⁻¹

(

h

)[

_h^Y

]) ◦ (

g∈^V

χ

⁻¹

(

g

)[

^Y_g

]) ∈

CH^dimY

(

Y

×

Y

)

_C

.

ByStep3,wealsoknowthat

h∈H

χ

⁻¹

(

h

)[

^Y_h

] =

⁰

∈

^CHdimY

(

Y

×

^Y

)

_C. 2

WearenowreadytoproveTheorem 1.2.Let

χ ∈

^{G∗ bea}non-trivialcharacter.Consider G

:=

^Ker

( χ :

^G

→ C

^∗

) ⊂

^G

.

Wetake X

:=

^X

/

Gwhichhasquotientsingularities(seeStep1intheproofofProposition 2.2),andwehaveafactorization f

:

X ^f

−→

X ^f

−→

A

.

Then f is acycliccoverwithGaloisgroup G

:=

^G

/

G

⊂

C^{∗.Moreover, thecharacter}

χ

is thepull-backofa non-trivial character

ι

ofG.ItisstraightforwardtoverifythattheassumptionofProposition 2.2issatisfiedby X andG.Hence

g∈^G

ι

⁻¹

(

g

)[

_g^X

] =

0

∈

CH^dimX

(

X

×

X

)

_C

.

Asbefore,

g∈G

χ

⁻¹

(

g

) [

_g^X

] = (

f

×

^f

)

^∗

(

g∈^G

ι

⁻¹

(

g

) [

_g^X

] ) =

⁰

∈

^CHdimX

(

X

×

^X

)

_C

,

andthus

[

_χ^X

] =

¹

|

^G

|

g∈^G

χ

⁻¹

(

g

)[

g^X

] =

⁰

∈

^CHdimX

(

X

×

^X

)

_C

. 2

Acknowledgements

WethankBarbaraFantechi,LieFu,JohannesSchmitt,andZhiyuTianforhelpfuldiscussions.

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