Quantum D-modules for toric nef complete intersections

(1)

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Quantum D-modules for toric nef complete intersections

Etienne Mann, Thierry Mignon

To cite this version:

Etienne Mann, Thierry Mignon. Quantum D-modules for toric nef complete intersections. 2011.

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(2)

QUANTUM

D

-MODULES FOR TORIC NEF COMPLETE INTERSECTIONS

by

Etienne Mann & Thierry Mignon

Abstra t. Onasmoothproje tivevarietywith

k

ampleline bundles,wedenoteby

Z

the omplete interse tionsubvarietydened bygeneri se tions.

Wedenethetwistedquantum

D

-modulewhi hisave torbundlewithaat onne tion,a atpairing andanaturalintegrablestru ture. An appropriatequotientof itisisomorphi to theambientpartofthequantum

D

-moduleof

Z

.

When the variety is tori , these quantum

D

-modules are y li . The twisted quantum

D

-module anbepresentedviamirrorsymmetrybytheGKZsystemasso iatedtothetotalspa e of thedualof thedire t sumofthese linebundles.

A questionis to know whatis thesystemof equations that denethe ambiantpart ofthe quantum

D

-moduleof

Z

. We onstru tthissystemasaquotientidealoftheGKZsystem.

Wealsostateandprovethenon-equivarianttwistedGromov-Wittenaxiomsintheappendix.

Contents

1. Introdu tion... 1

2. Twisted and redu ed quantum

D

-moduleswith geometri interpretation. 4 3. Batyrev rings fortori varieties with a splittedve tor bundle... 17

4. GKZ systems, quotient ideals and residual

D

-modules... 29

5. Isomorphisms between quantum

D

-modules and GKZ systems via mirror symmetry... 39

A. Twisted Axioms forGromov-Witten invariants... 48

B. Proof of Proposition 2.17... 54

Referen es... 58

1. Introdu tion

Mirrorsymmetryleadstomanydierentformulationsinmathemati s: equivalen eof

de-rived ategories(known asHomologi alMirror Symmetryby Kontsevi h [Kon95℄),

isomor-phism of Frobenius manifolds (see [Bar00℄), omparison of Hodge numbers for Calabi-Yau

varieties (see for example[Bat94℄), isomorphism of Givental's ones (see [Giv98℄),

isomor-phism of pure polarized TERP stru tures (see [Her06℄) or variation of non- ommutative

Hodge stru tures (see [KKP08℄), ...

2010 Mathemati s Subje t Classi ation. 14N35,53D45,14F10.

Key words and phrases. Quantum dierential modules, Gromov-Witten invariants, Batyrev rings, GKZsystems,mirrorsymmetry,

D

-modules,tori geometry.

E.MissupportedbythegrantoftheAgen eNationaledelaRe her heNewsymmetriesonGromov-Witten theories ANR-09-JCJC-0104-01.

(3)

look at quantum ohomology with a dierential module approa h : see Kim [Kim99℄ for

homogeneous spa es, see Coates-Corti-Lee-Tseng [CCLT06℄ and Guest-Sakai [GS08℄ for

weighted proje tive spa es,see alsothe worksof Iritani[Iri06℄,[Iri07℄,[Iri08℄ and[Iri09℄,

the book of Cox-Katz [CK99℄ and the one of Guest [Gue10℄.

From the small quantum produ t on a smooth proje tive variety, we an dene a trivial

ve torbundleover

H

0 _{(X, C)}

_{× V × C}

where

V

⊂ (C

∗

₎

r

and

r := dim

C

H

2 _{(X, C)}

whose ber is

H

∗

_{(X, C)}

. This bundle isendowed with aat onne tion and anon-degenerated pairing.

This onne tion is sometimes alled the Dubrovin-Givental onne tion. When

X

is a tori smooth Fano variety, Givental (see also Iritani [Iri09℄ for tori weak Fano orbifolds) gives

an expli it presentation of this

D

-module using GKZ systems. To prove this isomorphism, he uses the equality, up to amirror map, between the so alled

I

and

J

fun tions.

Intheveryni earti le[Iri09℄,Iritanienri hesthisquantum

D

-modulebyaddinganatural integral stru ture i.e., he denes a

Z

-lo alsystem whi h is ompatible with the onne tion. We all quantum

D

-module, denoted by

QDM(X)

, the trivial bundle endowed with a at onne tion, a at non-degenerated pairing and a natural integral stru ture. This

Z

-lo al system is natural in the following sense. Assume that

X

has a mirror (for instan e

X

is a weakFanotori orbifolds)thatisaLaurentpolynomialsu hthatitsBrieskornlatti e(whi h

isave tor bundlewithaat onne tion)isisomorphi tothe quantum

D

-moduleof

X

. On this Brieskorn latti e, we have a natural integral stru ture that omes fromthe Lefs hetz's

thimbles. The integral stru ture dened by Iritani is natural be ause it orresponds to

the naturalone on the mirror. Noti e that the bundle, the onne tion, the pairing and the

integralstru tureispartofthedenitionofaTERPstru turedenedbyHertlingin[Her06℄

or a variation of non- ommutative Hodge stru ture dened by Kontsevi h, Katzarkov and

Pantev in [KKP08℄.

In this paper, we investigate the same kind of obje ts asso iated to a smooth proje tive

variety

X

together with a splittedve tor bundle

E

whi h isglobally generated.

WeusethetwistedGromov-Witteninvariantsandthetwistedquantumprodu tstodene

a trivial ve tor bundle, denoted by

F

, on

H

0 _{(X, C)}

_{× V × C}

where

V

is an open in

(C

∗

₎

r

where the twisted quantum produ t is onvergent. Inspired by the lassi al ase, we dene

a at onne tion

∇

, a at pairing

S

and an integral stru ture

F

Z

on it. We all twisted quantum

D

-module,thequadruple

QDM(X,

E) := (F, ∇, S, F

Z

)

. Itsatisesalltheproperties of the lassi al

QDM(X)

ex ept that the pairing

S

is degenerated. We quotient by the kernel of

S

and we get a better obje t, alled redu ed quantum

D

-module and denoted by

QDM(X,

E) := (F , ∇, S, F

Z

)

. More pre isely, we onsider the trivial ve tor bundle

F

with thebers

H

2∗

_{(X, C)/ ker m}

c

top

where

m

c

top

: α

→ c

top

(

E)∪α

forany ohomology lass

α

. The data

(F,

∇, S, F

Z

)

pass tothisquotientand weget

QDM(X,

E)

that satisesallthe lassi al properties and now

S

is non-degenerated. So it really looks like a quantum

D

-module of a variety. Indeed, we have ageometri interpretation of

QDM(X,

E)

:

Theorem 1.1 (See Theorem 2.42). Let

L

1 , . . . ,

L

k

be ample line bundles on

X

, and assume that

dim

C

X

≥ k + 3

. Let

Z

be the zero of a generi se tion of

E := ⊕

k

i=1

L

i

. Denote by

ι : Z ֒

→ X

the losedembedding. Thenthe redu edquantum

D

-module

QDM(X,

E)

is iso-morphi to the sub-quantum

D

-module

QDM

amb

(Z)

of

QDM(Z)

whose ber is

ι

∗

_H

2∗

_{(X, C)}

.

Noti e thatour integral stru ture

F

Z

dened on

QDM(X,

E)

is naturalbe ause itindu es the natural one on

QDM

amb

(Z)

.

Then the next natural question is : an we nd a presentation of

QDM(X,

E)

and

QDM(X,

E)

when

X

is a tori smooth variety interms of GKZ systems ?

Denote by

D

the sheaf of dierential operators on the basis spa e of the

F

(this is not reallytrue, the operatorsthat we onsider are

zq

a

∂

q

a

where

q

a

are variable in

H

2 _{(X, C)}

(4)

z

isthe oordinateon

C

). Denote by

Y

the totalspa eofthe dualve tor bundle

E

∨

. Denote

by

G

the ideal sheaf asso iated to the GKZ system of the tori variety

Y

. We have the followingresult.

Theorem 1.2 (see Theorem 5.10). Let

X

beasmoothtori varietywith

k

linebundles

L

1 , . . . ,

L

k

su h that

(ω

X

⊗ L

1 ⊗ . . . ⊗ L

k

)

∨

is nef. We put

E := ⊕

k

i=1

L

i

.

1. If the line bundles

L

1 , . . . ,

L

k

are globally generatedthen we have the following isomor-phism :

D/G

−→ Mir

∼

∗

₍

_{F, ∇)}

where

Mir

isthe mirror map of Givental and

F

is the sheaf of se tions of

F

.

2. If the line bundles

L

1 , . . . ,

L

k

are ample, we have the following ommutative diagram

D/G

∼

Mir

∗

(

F, ∇)

D/ Quot(bc

top

,

G)

∼

Mir

∗

(

F, ∇)

where

bc

top

is an operator atta h to the ohomology lass

c

top

(

E)

( f.Notation 4.1) and

Quot(bc

top

,

G)

is the left quotient ideal

hP ∈ D, bc

top

P

∈ Gi

.

Noti e that, unlike the ommutative ase, the set

{P ∈ D, bc

top

P

∈ G}

is not an ideal. The ideal sheaf

Quot(bc

top

,

G)

answer to the following question whi h is addressed in the [CK99, p.94-95and p.101℄: What dierentialequationsshall we addto

G

toget an isomor-phism with

QDM

amb

(Z)

?

The isomorphisms above are based on the equality (up to the mirror map) between the

twisted

J

-fun tion and the twisted

I

-fun tionof Givental(see [Giv96℄and [Giv98℄)and a areful analysis of the lo alfreeness and rankof GKZ modules. Freeness and rankrequires

the study of Batyrev rings of the tori variety

Y

the total spa e of

E

∨

whi h willappear

as the restri tion of the

D

-modules at

z = 0

, and an be thought asa twisted Batyrev ring of the pair

(X,

E)

.

Proving this theorem leads to develop quite a lot of materialsand results whi h deserve

some pre isions. Letus sket h our strategy of proof.

Forthe rst pointof the theorem above,we show that

D/G

is alo allyfreesheaf of rank

dim

C

H

2∗

(X, C) = rk F

(see Theorem 4.10). This is done in

2

steps.

Werst prove the oheren e of

D/G

(see Theorem 4.5). This impliesthe lo alfreeness over

z

6= 0

and we use Adolphson's result in[Ado94℄ to ompute the rank.

On

z = 0

,we have atautologi al isomorphismbetween

D/G |

z=0

and the Batyrev ring of

Y

. We prove that this ring is lo ally free of rank

rk F

over a suitable algebrai neighborhood

U

(see below).

Forse ond point of the theorem above, we showin Theorem 4.14 :

On

z = 0

, we prove that the natural morphism between

D/ Quot(bc

top

,

G) |

z=0

and the residualBatyrev ring (see Denition 3.39) of

Y

is anisomorphism. We prove that this residual ring islo allyfree ofrank

rk F = dim H

2∗

_(X)

_{− dim ker m}

c

top

over

U

.

on

z

6= 0

the oheren e of

D/G

implies that

D/ Quot(bc

top

,

G)

islo allyfreeof rank less than

rk F

.

Letus olle tthe pre iseresultsthatwe proveonthe Batyrevrings,whi hare interesting

ontheir own :

Theorem 1.3. Let

X

be a smooth tori variety with

k

globally generated line bundles

L

1 , . . . ,

L

k

su hthatthe total spa e of the ve tor bundle

E := ⊕

k

(5)

divisor. Denote by

U

the good neighborhood in the spe trum of the Novikov ring dened in Notation 3.34.

1. (See Theorem 3.26) Denote by

B

the Batyrev ring (see Denition 3.12) of the total spa e of

E

∨

.The morphism :

Spec(B)

|

U

−→ U

is nite, at, of degree

dim H

2∗

_{(X, C)}

.

2. (See Proposition 3.40) Moreover, if the line bundles

L

1 , . . . ,

L

k

are ample then the morphism :

Spec(B

res

₎

_|

U

−→ U

is nite, at, of degree

dim H

2∗

(X, C)

where

B

res

is

the residualBatyrev ring (see Denition 3.39).

The plan of this arti le is the following.

In Se tion 2,we dene rst (Subse tion 2.1) the twisted quantum

D

-module

QDM(X,

E)

withallitspropertiesanditsnaturalintegralstru ture. Thenwedenetheredu edquantum

D

-module

QDM(X,

E)

in Subse tion 2.2. Finally, we give the geometri interpretation in Subse tion 2.3where we prove the rst Theorem 1.1.

In Se tion 3, we fo us on Batyrev rings for tori varieties. Noti e that this se tion an

be read independently of the rest of the paper. The rst Subse tion 3.1is devoted tosome

re alls on tori geometry. In Subse tion 3.1 we onstru t the fan of the total spa e of the

ve tor bundle

E

. In Subse tion 3.2, we dene the Batyrev rings. Subse tion 3.3 is devoted to some re alls on the primitive olle tions. In Subse tion 3.4, we prove that the quantum

Stanley-Reisner ideal has a Groebner basis indexed by primitive olle tions (See Theorem

3.22). In Subse tions 3.5and 3.6, we prove the Theorem 3.26 and Proposition 3.40 quoted

above inTheorem 1.3.

In Se tion 4, we fo us on GKZ modules. We prove rst that the GKZ module

D/G

is oherentinTheorem 4.5. Thenweprovethatitislo allyfreeof rank

rk F

inTheorem4.10. We nish by a result on the residual GKZ module

D/ Quot(bc

top

,

G)

(see Theorem 4.14). These results use Theorem 3.26 and Proposition 3.40 of the previous se tion.

InSe tion5,westartbysomere allonGivental'smirrorsymmetryinSubse tion5.1then

we state and prove Theorem 1.2 inSubse tion 5.2.

We nish this paper by two appendi es. Appendix A ontains the proof of the twisted

Gromov-Witten invariants in genus

0

that are known from the experts. We add it by la k of referen es.

Appendix B is a omplete proof of the atness of the onne tion

∇

using the twisted axioms.

A knowledgment: WethankThomasRei helt,ClaudeSabbahandChristianSevenhe k

for useful dis ussions. The seminar in Paris organized by Serguei Barannikov and Claude

Sabbahon the non- ommutativeHodge stru tures wasthe starting point of this paper. We

also thank Antoine Douai for helping in the organization of the workshop in Luminy on

the work of Iritani. We are also grateful to Hiroshi Iritani that pointed out the referen e

[Mav00 ℄ (see Remark 2.40) The rst author is supported by the ANR New symetries in

Gromov-Witten theories number ANR- 09-JCJC-0104-01.

Notation 1.4. We use alligraphi letters for the sheaves like

M, M

res

_,

_{G, B, L, E}

. We

use boldlettersfor modules or ideals onnon ommutativerings

M, M

res

_{, G, A, . . .}

.

2. Twisted and redu ed quantum

D

-modules with geometri interpretation

Let

X

be a smooth proje tive omplex variety of dimension

n

and

k

globally generated linebundles

L

1 , . . . ,

L

k

. Denote by

E

the sum

E := L

1 ⊕ · · · ⊕ L

k

.

Werstdenethetwistedquantum

D

-module,denotedby

QDM(X,

E)

,asso iatedtothese data (Denition 2.24). This is a trivial bundle of rank

dim

C

H

2∗

_{(X, C)}

with an integrable

(6)

It turns out that the pairing of the twisted quantum

D

-module is degenerated, whi h makes

QDM(X,

E)

a not so natural obje t, without lear geometri meaning. In a se ond paragraph we introdu e the redu ed quantum

D

-module

QDM(X,

E)

(Denition2.34) ; itis onstru ted as the quotient of

QDM(X,

E)

by the kernel of the endomorphism

m

c

top

, whi h is the up multipli ation by the Euler lass

c

top

(

E)

of

E

:

m

c

top

: H

2∗

(X, C)

−→ H

2∗

(X, C)

α

_{7−→ α ∪ c}

top

(

E).

The redu ed quantum

D

-module is a trivial bundle of rank

dim H

2∗

_{(X, C)}

_{− dim ker m}

c

top

with anintegrable onne tion,a at nondegenerated pairingand an integral stru ture.

If

dim X

≥ k + 3

, we also onsider a generi se tion of

E

and denote by

Z

the omplete interse tion subvariety dened as the zero lo us of this se tion. By Bertini's theorem over

C

, the subvariety

Z

is smooth and onne ted. Assuming moreover that the

L

i

are ample linebundles, the Lefs hetz theorem givesan isomorphismbetween

H

2 _{(X, C)}

and

H

2 _{(Z, C)}

.

We an ompare

QDM(X,

E)

,

QDM(X,

E)

and the lassi al, untwisted, quantum

D

-module of

Z

,

QDM(Z)

. This willbe made inthe last subse tion.

Notation 2.1. For

0 ≤ i ≤ 2n

, denote by

H

i

_{(X) := H}

i

_{(X, C)}

the omplex

ohomol-ogy group of lasses of degree

i

. Also denote by

H

∗

_(X)

the omplex ohomology ring

⊕

2n

i=0

H

i

(X)

; the even part of this ring will be written

H

2∗

_(X)

. Put

s = dim

C

H

2∗

_(X)

and

r = dim

C

H

2 (X)

.

Wex,on e andforall,ahomogeneousbasis

(T

0 , . . . , T

s−1

)

of

H

2∗

_(X)

su hthat

T

0 = 1

is the unitforthe up produ t andthatthe lasses

T

1 , . . . , T

r

formabasis of

H

2 _{(X, Z)}

modulo

torsion. Denoteby

(t

0 , . . . , t

s−1

)

theasso iated oordinatesandput

τ :=

P

_s−1

a=0

t

a

T

a

and

τ

2 :=

P

r

a=1

t

a

T

a

. Also denoteby

(T

0 _{, . . . , T}

s−1

₎

the Poin aré dual in

H

2∗

_(X)

of

(T

0 , . . . , T

s−1

)

. Asa onvention, Wewillwrite

H

2 (X, Z)

forthedegree

2

integerhomologymodulotorsion. Denoteby

(B

1 , . . . , B

r

)

thedualbasisof

(T

1 , . . . , T

r

)

in

H

2 (X, Z)

. Theasso iated oordinates willbe denoted by

(d

1 , . . . , d

r

)

.

We denote by

T

X

the tangent bundle of

X

,

ω

X

the anoni al sheaf, and x a anoni al divisor

K

X

.

Asa onvention,wewillmakenonotationaldistin tionbetweenve torbundlesandlo ally

free sheaves, writingfor example

E

and

L

i

forboth.

2.1. Twistedquantum

D

-module. Inthissubse tion,wedenethetwistedquantum

D

-module

QDM(X,

E) = (F, ∇, S, F

Z

)

.

2.1.a. Twistedquantumprodu t. Firstre allthedenitionofthetwistedGromov-Witten

invariant ( f.[Giv96℄and [CG07℄ or[CK99, Se tion 11.2.1℄and [Pan98℄).

Let

ℓ

be in

N

and

d

be in

H

2 (X, Z)

. Denote by

X

0,ℓ,d

the modulispa e of stable maps of degree

d

fromrational urveswith

ℓ

marked points to

X

. The universal urve over

X

0,ℓ,d

is

X

0,ℓ+1,d

:

X

0,ℓ+1,d

π

e

ℓ+1

X

0,ℓ,d

where

π

isthemap thatforgets the

(ℓ + 1)

-thpointandstabilizes,and

e

ℓ+1

isthe evaluation atthe

(ℓ + 1)

-thmarked point.

Re all that a onvex bundle

N

on

X

is a ve tor bundle su h that, for any stable map

f : C

→ X

where

C

is a rationalnodal urve,

H

1 _{(C, f}

∗

_{N ) = 0}

(7)

Proposition 2.2. Let

N

beagloballygeneratedve torbundle(notne essarilysplitted)of rank

b

then

N

is onvexandthesheaf

N

0,ℓ,d

:= R

0 _π

∗

e

∗

ℓ+1

N

islo allyfreeofrank

R

d

c

1 (

N )+b

. Proof. Let usprove the onvexity of

N

. Wefollow[FP97 ,Lemma 10℄.

Let

f : C

−→ X

be a stable map and

p

be a non singular pointon

C

. We willprove by indu tion onthe number of irredu ible omponents of

C

that

H

1 _{(C, f}

∗

_{N ⊗ O}

C

(

−p)) = 0.

(2.3)

First,assumethat

C

≃ P

1

. We anwrite

f

∗

₍

_{N ) ≃ ⊕}

b

i=1

O

P

1 (a

i

)

with

a

1 , . . . , a

b

in

Z

. Sin e

N

isgloballygenerated,

f

∗

₍

_{N )}

alsois,whi himplies

a

i

≥ 0

forany

i

in

{1, . . . , b}

. Itfollows that

H

1 _(P

1 _{, f}

∗

_{N ⊗ O}

C

(

−p)) = ⊕

b

i=1

H

1 (P

1 ,

O

P

1 (a

i

− 1)) = 0

. Assumenowthat

C = C

′

_{∪ C}

0

where

C

0 ≃ P

1

and

p

in

C

0

. Denote by

p

1 , . . . , p

q

thepoints of

C

0 ∩ C

′

. Noti e that

C

′

has exa tly

q

onne ted omponents interse ting

C

0

on exa tly one point. Ea h

p

i

is a smooth point of one of these omponents. We have the following exa t sequen e

0 f

∗

_{N ⊗ O}

C

′

(

−

P

q

i=1

p

i

)

f

∗

N ⊗ O

C

(

−p)

f

∗

N ⊗ O

C

0 (

−p)

0

From the asso iatedlong exa t sequen e and by the indu tive assumptiononthe onne ted

omponents of

C

′

,we dedu e the equality (2.3). The exa t sequen e :

0 f

∗

_{N ⊗ O}

C

(

−p)

f

∗

N

f

∗

N ⊗ O

p

0

gives

H

1 _{(C, f}

∗

_{N ) = 0}

.

Now, the stalk at a point

(C, x

1 , . . . , x

ℓ

, f : C

→ X)

in

X

0,ℓ,d

of the K-theoreti push-forward

N

0,ℓ,d

is

H

0 _{(C, f}

∗

_{N ) − H}

1 _{(C, f}

∗

_{N )}

. Sin e

N

is onvex

H

1 _{(C, f}

∗

_{N ) = 0}

and

H

0 _{(C, f}

∗

_{N )}

has dimension

R

d

c

1 (

N ) + b

by Riemann-Ro h. Thus,

N

0,ℓ,d

is lo ally free of dimension

R

d

c

1 (

N ) + b

on

X

0,ℓ,d

. Let

E

0,ℓ,d

bethe sheaf

R

0 _π

∗

e

∗

ℓ+1

E

asin Proposition 2.2. For

j

in

{1, . . . , ℓ}

,we dene the surje tive morphism

E

0,ℓ,d

→ e

∗

j

E

by evaluating the se tion to the

j

-th marked point. We dene

E

0,ℓ,d

(j)

tobethe kernel of this map that is we have the following exa t sequen e

0 E

0,ℓ,d

(j)

E

0,ℓ,d

e

∗

j

E

0

(2.4)

By Proposition 2.2, for any

j

∈ {1, . . . , ℓ}

the bundle

E

0,ℓ,d

(j)

has rank

R

d

c

1 (

E)

.

For

i

∈ {1, . . . , ℓ}

, let

N

i

bethe linebundleon

X

0,ℓ,d

whoseber atapoint

(C, x

1 , . . . , x

ℓ

,

f : C

→ X)

isthe otangent spa e

T

∗

_C

x

i

. Put

ψ

i

:= c

1 (N

i

)

in

H

2 _(X

0,ℓ,d

)

. Denition 2.5. Let

ℓ

be in

N

,

γ

1 , . . . , γ

ℓ

be lasses in

H

2∗

_(X)

,

d

be in

H

2 (X, Z)

and

(m

1 , . . . , m

ℓ

)

be in

N

ℓ

. For

j

in

{1, . . . , ℓ}

, the (

j

-th) twisted Gromov-Witten invariant with des endants of these data is dened and denoted by

D

τ

m

1 (γ

1 ), . . . , ^

τ

m

j

(γ

j

), . . . , τ

m

ℓ

(γ

ℓ

)

E

0,ℓ,d

:=

Z

[X

0,ℓ,d

]

vir

c

top

(

E

0,ℓ,d

(j))

ℓ

Y

i=1

ψ

m

i

e

∗

i

γ

i

where

e

i

: X

0,ℓ,d

→ X (1 ≤ i ≤ ℓ)

is the evaluationmorphism to the

i

th marked point and

[X

0,ℓ,d

]

vir

is the virtual lass on

X

0,ℓ,d

. Denition 2.6. Let

τ

2

bea lass of

H

2 _(X)

and

γ

1 , γ

2

bein

H

2∗

_(X)

. The twisted small

quantum produ t (withrespe t to

E

) of

γ

1

and

γ

2

is dened by

γ

1

• tw

τ

2 γ

2 :=

s−1

X

a=0

X

d∈H

2 (X,Z)

e

R

d

τ

2 D

_γ

1 , γ

2 , e

T

a

E

0,3,d

T

a

(8)

τ

in

H

2∗

_(X)

, abig twisted quantum produ t :

γ

1

• tw

τ

γ

2 :=

s−1

X

a=0

DD

γ

1 , γ

2 , e

T

a

EE

0 T

a

_.

As usual, wehave :

• tw

τ

2 :=

• tw

τ

|

τ =τ

2

. Wewillnot use of bigtwisted quantum produ ts. 2. One an alsodene the smalltwistedquantum produ t without hoosing a basis by :

γ

1

• tw

τ

2 γ

2 :=

X

d∈H

2 (X,Z)

e

R

d

τ

2 e

3∗

e

∗

1 γ

1 ∪ e

∗

2 γ

2 ∪ c

top

(

E

0,3,d

(3))

∩ [X

0,3,d

]

vir

(2.8)

2.1.b. Parameters. The quantum produ t written in Denition 2.6 depends on the

pa-rameter

τ

2

in

H

2 _(X)

. The Pi ard group

Pic(X)

a ts on

H

2 _(X)

inthe followingway : for

L

in

Pic(X)

,

L.τ

2 = τ

2 + 2

√

−1πc

1 (

L)

. The number

e

R

d

τ

2

being invariant by this a tion, the quantum produ t is naturallydened over

H

2 _{(X)/ Pic(X) = H}

2 _(X)/2

√

_−1πH

2 _{(X, Z)}

.

Letus extend the lo us of the parameter. Denote by NE

(X)

Z

⊂ H

2 (X, Z)

the Mori one of

X

, generated asa semi-group by numeri al lasses of irredu ible urves in

X

.

Notation 2.9. The semigroup algebras of NE

(X)

Z

and

H

2 (X, Z)

will be respe tively denoted by

Λ

and

Π

:

Λ = C[

NE

(X)

Z

] = C[Q

d

_{, d}

_∈

NE

(X)

Z

],

Π = C[H

2 (X, Z)] = C[Q

d

_{, d}

_{∈ H}

2 (X, Z)]

where

Q

d

are indeterminates satisfyingrelations :

Q

d

_.Q

d

′

= Q

d+d

′

.

The s heme

Spec Λ

is an irredu ible, possibly singular, ane variety of dimension

r

. De-note by

V

the set of omplex points of

Spec Λ

. Points of

V

are hara ters

(1)

of NE

(X)

Z

. If

q

is su h a hara ter, denote by

q

d

its evaluation on

d

in NE

(X)

Z

. Sin e

X

is pro je -tive, the Mori one is stri tly onvex and there exists a unique hara ter sending any

d

in

H

2 (X, Z)

\ {0}

to

0

. Wewilldenotethis hara ter by

0

and allit,asusual,the large radius limit of

X

.

The s heme

Spec Π

is atorus ofdimension

r = rk H

2 (X, Z)

. The set of omplex pointsof

Spec Π

will be denoted by

T

; a point of

T

is a hara ter of

H

2 (X, Z)

and

T

is a smooth subset of

V

. We will identify

T

and

H

2 _(X)/2

√

_−1πH

2 _{(X, Z)}

via the natural surje tive

morphismof omplex variety :

Ψ : H

2 (X, C)

−→ T

(2.10)

τ

_{7−→ q}

τ

:=

h

d

_{∈ H}

2 (X, Z)

7→ q

τ

d

= e

R

d

τ

i

The kernel of

Ψ

is

2 √

−1πH

2 _{(X, Z)}

. Thus, the large radius limit

0

in

V

⊃ T

is a limit in

H

2 _(X)/2

√

_−1πH

2 _{(X, Z)}

.

The smallquantum produ t an now be dened with parameter

q

in

V

:

Denition 2.11. Let

q

be in

V

and

γ

1 , γ

2

be in

H

2∗

_(X)

. The twisted small quantum

produ t is dened by

γ

1

• tw

q

γ

2 :=

s−1

X

a=0

X

d∈H

2 (X,Z)

q

d

D

γ

1 , γ

2 , e

T

a

E

0,3,d

T

a

(1)

Bya hara ter ofasemi-group

R

of

H

2 (X, Z)

wemean anappli ation

q

: R

−→ C

su h that

q(0) = 1

and

q(d + d

′

_{) = q(d).q(d}

′

₎

forany

d, d

′

in

R

. If

R

isagrouptheimageof

q

isin

C

∗

. If

q

issu ha hara ter, wewillwrite

q

d

_{:= q(d)}

. A hara ter

q

ofasemi-group

R

givesa omplexpoint

Spec C

−→ Spec C[R]

whi h will alsobedenotedby

q

; this orresponden eisabije tion. Noti ethat,if

d

is in

R

,

Q

d

isafun tion on

Spec

C

C[R]

andwehave:

Q

d

_{(q) = q}

d

.

(9)

Denition2.11andDenition2.6are ompatible: Forany

τ

2

in

H

2 _(X)

,

Ψ(τ

2 )

isin

T

and

γ

1

• tw

τ

2 γ

2 = γ

1

• tw

Ψ(τ

2 )

γ

2

.

Assumption 2.12. We willassumethat there exists an open subset

V

of

V

ontaining the large radius limit

0

su h that :

∀q ∈ V , ∀γ

1 , γ

2 ∈ H

2∗

(X), γ

1

• tw

q

γ

2

is onvergent.

This assumptionis easilyshown tobe true whenthe linebundle

(ω

X

⊗ L

1 ⊗ · · · ⊗ L

k

)

∨

is

ample,thatiswhenthe ompleteinterse tionvariety

Z

dened by

E

isFano. Inother ases, su h as Calabi-Yausubvarieties of tori varieties onsidered below, one may use [Iri07℄ to

he k this assumption.

Notation 2.13. We denote by

V

the omplex nonsingular variety

V := V

∩ T

.

Thus,

V

is a smooth lo us in

V

where the quantum produ t is onvergent. Wehave : large radius limit

= 0

∈ V

( onvergent produ t)

⊂

V

= Spec

C

Λ

∪

0 /

∈ V

( onvergent produ t)

⊂ T = Spec

C

Π

∼

−→ (C

∗

₎

r

As a onvention, we willdenote neighborhood of

0

in

V

by an overlined apital letter, and itsinterse tionwith

T

by thesame apitalletterwithoutoverlining(

V

isa ompa ti ation of

T

inthe neighbourhood of the large radiuslimit).

Letus re all some properties of the twisted quantum produ t :

Proposition 2.14. For any

q

in

V

the twisted quantum produ t

• tw

q

isasso iative, om-mutative, with unity

T

0 := 1

.

Proof. This isa lassi alproof, assoonasthe twistedGromov-Wittenaxiomsare known.

The twisted axioms are shown in Appendix A. Su h proves are given by Pandharipande in

[Pan98℄, Proposition 3, for a smooth hypersurfa e of

P

n

and by Iritani in Remark 2.2. of

[Iri11℄, inthe general ase.

2.1. . Thetrivialbundlewithanintegrable onnexion. Usingbasis

T

1 , . . . , T

r

and

B

1 , . . . ,

B

r

denedin2.1,wehave:

Π = C[H

2 (X, Z)]

∼

−→ C[q

±

1 , . . . , q

r

±

]

where

q

a

:= Q

B

a

( f.footnote 1). Thus if

d =

P

r

a=1

d

a

B

a

we get

Q

d

₌

Q

r

a=1

q

a

d

a

in

Π

. Viewing the

q

a

's as oordinates of

T

, we get :

q

d

₌

Q

r

a=1

q

a

d

a

for any

q

∈ T

. For

a

in

{1, . . . , r}

, we put :

δ

a

:= q

a

∂

q

a

δ

z

:= z∂

z

.

Re all that

t

0

is the oordinateon

H

0 _(X)

.

Notation 2.15. We denote by

F

the trivialholomorphi ve tor bundleof ber

H

2∗

_(X)

over

H

0 _(X)

_{× V × C}

:

F :=

H

2∗

(X)

× H

0 _(X)

_{× V × C}

_{→ H}

0 _(X)

_{× V × C}

together with the following meromorphi onne tion :

∇

∂

_t0

:= ∂

t

0 +

1 z

1

• tw

q

,

∇

δa

:= δ

a

+

1 z

T

a

• tw

q

,

∇

δ

z

:= δ

z

−

1 z

E

• tw

q

+µ

where

µ

isthe diagonalmorphismdened by

µ(T

a

) :=

1

2 (deg(T

a

)

− (dim

C

X

− rk E)) T

a

and

E

_(t

₀

_{, q, z) := t}

₀

1 _{+ c}

₁

₍

_T

_X

_{⊗ E}

∨

₎

. This global se tion

E

of

F

orresponds to the Euler eld. Noti ethatthetwistedprodu t

• tw

q

doesnotdependon

t

0

be auseofthetwistedfundamental lass Axiom ( f.PropositionA.4).

(10)

In the untwisted ase, it is known that

∇

is aat onne tion and its at se tions an be des ribed expli itly. Letus give the equivalent property in the twisted ase. We dene the

multi-valuated ohomologi almeromorphi fun tion

L

tw

_(t

0 , q, z) :

H

2∗

(X)

_{−→ H}

2∗

(X)

γ

7−→ L

tw

_(t

0 , q, z)γ = e

−t

0 /z





q

−T /z

γ

−

s−1

X

a=0

X

H2(X,Z)

d6=0

q

d

q

−T /z

_γ

z + ψ

, e

T

a

0,2,d

T

a







(2.16) where

ψ := ψ

1 = c

1 (N

1 )

isthe lass of

H

2 _(X

0,3,d

)

given beforeDenition 2.5,

1 z + ψ

:=

X

k∈N

(

−1)

k

_ψ

k

_z

−k−1

_,

q

−T /z

= q

−T

1 /z

_.

_{· · · .q}

−T

r

/z

_{:= e}

−z

−1

P

r

a=1

T

a

log(q

a

)

and

log(q

a

)

isthe multi-valuatedfun tion,or any determinationof the logarithm ona simply onne ted open subset of

V .

For an endomorphism

u

, we denote

z

u

_{:= exp(u log z)}

. The following Proposition is the

twisted version of Proposition2.4in [Iri09℄.

Proposition 2.17. 1. The onne tion

∇

is at. 2. For

a

in

{1, . . . , r}

and

γ

∈ H

2∗

_(X)

we have

∇

∂

_t0

L

tw

(t

0 , q, z)γ = 0,

∇

δ

a

L

tw

_(t

0 , q, z)γ = 0

∇

δ

z

L

tw

_(t

0 , q, z)γ = L

tw

(t

0 , q, z)

µ

−

c

1 (

T

X

⊗ E

∨

₎

z

γ

3. The multi-valued ohomologi alfun tion

L

tw

_(t

0 , q, z)z

−µ

z

c

1 (T

X

⊗E

∨

₎

is a fundamental so-lution of

∇

above

H

0 _(X)

_{× V × C}

.

Noti e that, as afundamental solution,

L

tw

is onvergent above

H

0 _(X)

_{× V × C}

.

This kindof resultis lassi alinthe untwisted ase ([CK99℄, [Iri09℄). Byla k referen es

ontwistedGromov-Witten invariants,we write down a proof infull details inAppendix B.

2.1.d. The degenerated pairing. Denote by

(

·, ·)

the Poin aré duality on

H

2∗

_(X)

. As

D

γ

1 , γ

2 , e

T

a

E

0,3,d

isnot symmetri inthe three arguments wedo not havethe Frobenius

rela-tion, that is :

(γ

1

• tw

q

γ

2 , γ

3 )

6= (γ

1 , γ

2

• tw

q

γ

3 ).

Nevertheless we an dene asymmetri bilinearform :

Denition 2.18. The twisted pairingon

H

2∗

_(X)

isdened by :

∀γ

1 , γ

2 ∈ H

2∗

(X), (γ

1 , γ

2 )

tw

:=

Z

X

γ

1 ∪ γ

2 ∪ c

top

(

E).

Proposition 2.19. 1. The bilinear form

(

·, ·)

tw

is degenerated with kernel

ker m

c

top

where

m

c

top

is dene as :

m

c

top

: H

2∗

(X)

−→ H

2∗

(X)

(11)

2. For

γ

1 , γ

2 , γ

3

in

H

∗

_(X)

, we have the Frobenius relation :

(γ

1

• tw

q

γ

2 , γ

3 )

tw

= (γ

1 , γ

2

• tw

q

γ

3 )

tw

.

Proof. The rst laimis obvious.

By Denition 2.18 and Remark 2.8, it is enough to prove the following equality for any

d

_{∈ H}

2 (X, Z)

:

Z

X

e

_3∗

e

∗

₁

γ

1 ∪ e

∗

2 γ

2 ∪ c

top

(

E

0,3,d

(3))

∩ [X

0,3,d

]

vir

∪ γ

3 ∪ c

top

(

E)

=

Z

X

e

_3∗

e

∗

1 γ

2 ∪ e

∗

2 γ

3 ∪ c

top

(

E

0,3,d

(3))

∩ [X

0,3,d

]

vir

∪ γ

1 ∪ c

top

(

E).

The exa t sequen e :

0 E

0,3,d

(3)

E

0,3,d

e

∗

3 E

0

gives

c

top

(

E

0,3,d

(3)).c

top

(e

∗

3 E) = c

top

(

E

0,3,d

)

. By proje tion formulawe get :

Z

X

e

_3∗

e

∗

₁

γ

1 ∪ e

∗

2 γ

2 ∪ c

top

(

E

0,3,d

(3))

∩ [X

0,3,d

]

vir

∪ γ

3 ∪ c

top

(

E)

=

Z

[X

0,3,d

]

vir

e

∗

1 γ

1 ∪ e

∗

2 γ

2 ∪ e

∗

3 γ

3 ∪ c

top

(

E

0,3,d

)

As the lastnumberis invariant by permuting the lass

γ

i

, we dedu e the proposition. Let

O := O

H

0 (X)×V ×C

be the sheaf of holomorphi fun tions on

H

0 _(X)

_{× V × C}

, and

F

be thesheafof holomorphi se tionsof

F

. Let

Γ(

O)

be the ringof globalse tionsof

O

,and

Γ(

F)

be the

Γ(

O)

-modulesof global se tion of

F

;

Γ(

O)

is endowed with the involution :

κ :

Γ(

O)

−→

Γ(

O)

f (t

0 , q, z)

7−→ f

κ

:= f (t

0 , q,

−z)

Denote by

Γ(

F)

κ

the

Γ(

O)

-moduleequals,asaset, to

Γ(

F)

and endowed with thefollowing multipli ation :

∀f ∈ Γ(O), s ∈ Γ(F), f.s := f

κ

_.s

. Wedene a asesquilinear pairing

S : Γ(

F)

κ

_{⊗ Γ(F) −→ Γ(O)}

by xing itsvalue on onstant se tionsof

F

:

∀γ

1 , γ

2 ∈ H

2∗

(X), S(γ

1 , γ

2 ) = (γ

1 , γ

2 )

tw

.

As a onsequen e, weget :

∀s

1 , s

2 ∈ Γ(F), ∀(t

0 , q, z)

∈ H

0 (X)

× V × C,

S(s

1 , s

2 )(t

0 , q, z) = (s

1 (t

0 , q,

−z), s

2 (t

0 , q, z))

tw

.

Proposition 2.20. 1. The pairing

S

is

∇

-at. 2. For any

s

1 , s

2

in

Γ(

F)

,

S(L

tw

.s

1 , L

tw

.s

2 ) = S(s

1 , s

2 ).

3. For any

γ

1 , γ

2

in

H

2∗

_(X)

we have

S(L

tw

(t

0 , q, z)z

−µ

z

c

1 (T

X

⊗E

∨

₎

γ

1 , L

tw

(t

0 , q, z)z

−µ

z

c

1 (T

X

⊗E

∨

₎

γ

2 )

=S(e

√

−1πc

1 (T

X

⊗E

∨

)

_γ

1 , e

√

−1πµ

_γ

2 ).

(12)

Proof. 1. Bythe FrobeniuspropertyofProposition2.19.(2),forany

a

∈ {1, . . . , r}

and for any

s

1 , s

2 ∈ Γ(F)

, wehave :

δ

a

S(s

1 , s

2 ) = S(

∇

δ

a

s

1 , s

2 ) + S(s

1 ,

∇

δ

a

s

2 )

∂

t

0 S(s

1 , s

2 ) = S(

∇

∂

t0

s

1 , s

2 ) + S(s

1 ,

∇

∂

t0

s

2 ).

By the denition of

µ

and Proposition 2.19.(2), for any

s

1 , s

2 ∈ Γ(F)

we have

δ

z

S(s

1 , s

2 ) = S(

∇

δ

z

s

1 , s

2 ) + S(s

1 ,

∇

δ

z

s

2 ).

Hen e,

S

is

∇

-at.

2. By atnessof

S

and Proposition 2.17.(2), we dedu e that

∀γ

1 , γ

2 ∈ H

2∗

(X), δ

a

S(L

tw

γ

1 , L

tw

γ

2 ) = 0.

So the expression

(L

tw

_(t

0 , q,

−z)γ

1 , L

tw

(t

0 , q, z)γ

2 )

tw

does not depend on

q

. By the asymptoti of

L

tw

atthe large radius limit,we get

(L

tw

(t

0 , q,

−z)γ

1 , L

tw

(t

0 , q, z)γ

2 )

tw

∼

q=0

(q

−T /z

γ

1 , q

T /z

γ

2 )

tw

= (γ

1 , γ

2 )

tw

.

Therelation

S(L

tw

_.s

1 , L

tw

.s

2 ) = S(s

1 , s

2 )

isalsotrueforany

s

1 , s

2 ∈ Γ(F)

by sesquilin-earity.

3. By the previous formula and the atness, we dedu e that the left hand side does not

depends on

z

. Sowe an put

z = 1

. We dedu e that the lefthand side is equalto

S(e

−

√

−1πµ

e

√

−1πc

1 (T

X

⊗E

∨

)

_γ

1 , γ

2 ).

As

S(

−µ(γ

1 ), γ

2 ) = S(γ

1 , µ(γ

2 ))

for any

γ

1 , γ

2

in

H

2∗

_(X)

, we dedu e the formula.

2.1.e. Integral stru ture. In the same way than Iritani [Iri09, Denition 2.9℄ (see also

[Iri11, footnote 8 p.20℄), we dene an integral stru ture on the ve tor bundle

F

with on-ne tion

∇

, ompatible tothe pairing

S

.

Denote by

γ

the Euler onstant. Fora ve tor bundle

N

on

X

of rank

b

, we onsider the invertible ohomology lass

b

Γ(

_{N ) :=}

b

Y

i=1

Γ(1 + ν

i

) = exp

−γc

1 (

N ) +

X

b≥2

(

₋₁₎

b

(b

_{− 1)!ζ(b) Ch}

b

(

N )

!

where

ν

1 , . . . , ν

b

are the Chernrootsof

N

and

Ch

b

(

N )

is the lassof degree

2b

of the Chern hara ter

Ch(

N )

. Denote by

K(X)

the Grothendie k group of ve tor bundleson

X

. Re all that the morphism

Ch : K(X)

→ H

2∗

_{(X, Z)}

be ome an isomorphism after tensored by

C

(see for instan e Theorem 3.25 p.283in [Kar78℄).

Denition 2.21. Forany

v

in

K(X)

,we put

Z

tw

(v) := (2π)

−(n−k)/2

L

tw

(t

0 , q, z)z

−µ

z

c

1 (T

X

⊗E

∨

₎

b

Γ(

_T

X

)b

Γ(

E)

−1

(2

√

−1π)

deg /2

Ch(v).

We all

Z

tw

_(K(X))

the

b

Γ

-integral stru ture on

QDM(X,

E)

and wedenote it by

F

Z

. Remark 2.22. Noti ethat

Z

tw

_(v)

isamulti-valuedatse tionofthebundle

(F,

∇)

and that

Z

tw

_(K(X))

_⊗

Z

C

isthe set ofat se tionsof

F

. We an understandthe formulaof

Z

tw

above asthe twist by

b

Γ(

T

X

)b

Γ(

E)

−1

of the naturalintegral stru ture given by

K(X)

.

K(X)

(2

√

−1π)

deg /2

_Ch

(F, d)

b

Γ(T

X

)b

Γ(E)

−1

(F, d)

(2π)

−n/2

_L

tw

_(t

0 ,q,z)z

−µ

z

c1(TX ⊗E

∨ )

(F,

_∇)

v

1 , v

2

in

K(X)

, we have :

S(

Z

tw

_(v

1 ),

Z

tw

(v

2 )) =

Z

X

(13)

Proof. Using Proposition2.20.(3) and

e

√

−1πµ

_{= (}

₋₁₎

deg /2

₍

√

₋₁₎

k−n

, we dedu e that

S(

_Z

tw

(v

1 ),

Z

tw

(v

2 ))

=(2

√

_−1π)

k−n

Z

X

c

top

(

E)e

√

−1πc

1 (T

X

⊗E

∨

)

_Γ(

b

_T

X

)b

Γ(

E)

−1

(2

√

−1π)

deg /2

Ch(v

1 )

∪ (−1)

deg /2

_Γ(

_b

_T

X

)b

Γ(

E)

−1

(2

√

−1π)

deg /2

_Ch(v

2 )

We have the following fa ts : for any

α, β

in

H

2∗

_(X)

, for any

v

in

K(X)

and for any

δ

∈ H

2 _(X)

,

β

∪ (2

√

−1π)

deg /2

_{α = (2}

√

_−1π)

deg /2

_(β/(2

√

_−1π)

deg β/2

_{∪ α)}

Z

X

(2

√

−1π)

deg /2

_{α = (2}

√

_−1π)

n

Z

X

α

(

₋₁₎

deg /2

Γ(1 + δ) = Γ(1

_{− δ)(−1)}

deg /2

(

₋₁₎

deg /2

Ch(v) = Ch(v

∨

).

Denote by

ν

1 , . . . , ν

n

the Chern root of

T

X

and

ǫ

1 , . . . , ǫ

k

the Chern roots of

E

. From the above properties, we dedu e that

S(

_Z

tw

(v

1 ),

Z

tw

(v

2 )) =

Z

X

c

top

(

E)e

c

1 (T

X

⊗E

∨

_)/2

n

Y

i=1

Γ

1 +

ν

i

2 √

−1π

Γ

1 ₋

ν

i

2 √

−1π

∪

k

Y

j=1

Γ

1 +

ǫ

j

2 √

−1π

−1

Γ

1 ₋

ǫ

j

2 √

−1π

−1

Ch(v

1 ⊗ v

2 ∨

)

Usingthe formalidentity

Γ(z)Γ(1

− z) =

π

sin(πz)

, we dedu e that

Γ(1

_{− z)Γ(1 + z) =}

ze

−z/2

1 − e

−z

.

This implies the formula.

Re all fromDenition 2.21 that we denote

F

Z

the integral stru ture

Z

tw

_(K(X))

.

Denition 2.24. Thetwisted quantum

D

-moduledenotedby

QDM(X,

E)

isthe quadru-ple

(F,

∇, S, F

Z

)

.

2.2. Redu edquantum

D

-module. Inthissubse tionwedenetheredu edquantum

D

-module, denoted by

QDM(X,

E)

, whi h is a quadruple

F ,

∇, S, F

Z

. The pairing

S

is non-degenerated.

Re all that

m

c

top

isthe endomorphism

m

c

top

: H

2∗

(X)

−→ H

2∗

(X)

α

7−→ c

top

(

E) ∪ α.

Put

H

2∗

(X) := H

2∗

_{(X)/ ker m}

c

top

and all it the redu ed ohomology ring of

(X,

E)

. Sin e

m

c

top

is agraded morphism, the ve tor spa e

H

2∗

(X)

is naturally graded. For

γ

∈ H

2∗

_(X)

,

we denoteby

γ

its lass in

H

2∗

(X)

.

Denote by

F

the trivialbundle

H

2∗

(X)

× H

0 _(X)

_{× V × C → H}

0 _(X)

_{× V × C}

. On

F

,we willdene a onne tion

∇

and a non-degenerated paring

S

. They willbe indu ed by those on

F

.

For any

γ

1 , γ

2 ∈ H

2∗

_(X)

, dene the redu ed pairing

(

·, ·)

red

whi h is a bilinear form on

H

2∗

_(X)

by

(γ

1 , γ

2 )

red

:= (γ

1 , γ

2 )

tw

.

(14)

ByProposition2.19,

ker m

c

top

isthekernelofthe twistedpairing. Itfollows thattheredu ed pairing isa welldened and non degenerated bilinearform.

Wedene the pairing

S

as we did for

S

but hanging

(

·, ·)

tw

by

(

·, ·)

red

( f.before

Propo-sition 2.20). From (2.25),for any

s

1 , s

2 ∈ Γ(H

0 _(X)

_{× V × C, F)}

,we dedu e that

S(s

1 , s

2 ) = S(s

1 , s

2 )

(2.26)

Let

(φ

0 , . . . , φ

s

′

−1

)

be a homogeneousbasis of

H

2∗

(X)

and denote

(φ

0 _{, . . . , φ}

s

′

₋₁

)

its dual basis with respe t to

(

·, ·)

red

.

Denition 2.27. Let

γ

1 , . . . , γ

n

be lasses in

H

2∗

_(X)

.

1. Let

d

bein

H

2 (X, Z)

. The redu ed Gromov-Witten invariant is

hγ

1 , . . . , γ

n

i

red

0,ℓ,d

:=

hγ

1 , . . . , ^

c

top

(

E)γ

_n

i

0,ℓ,d

2. The redu ed quantum produ t is

γ

₁

_•

red

_q

γ

₂

:=

s−1

X

a=0

X

d∈H

2 (X,Z)

q

d

_hγ

₁

, γ

₂

, φ

a

i

red

0,3,d

φ

a

Remark 2.28. By the twisted

S

n

-symmetri axiom ( f.Proposition A.2), the redu ed Gromov-Witteninvariants are welldened onthe lass in

H

2∗

(X)

. Noti e that the redu ed Gromov-Witten invariant are

S

n

symmetri . The onvergen e domain of

• red

q

ontains

V

. Wewillrestri t ourselvesto

V

.

γ

1 , γ

2

in

H

2∗

_(X)

, we have

γ

1

• tw

q

γ

2 = γ

1

• red

q

γ

2

Proof. Using Formula(2.8) for the twisted quantum produ t we get :

γ

1

• tw

q

γ

2 =

X

d∈H

2 (X,Z)

q

d

e

_3∗

α

where we put

α := e

∗

1 γ

1 ∪ e

∗

2 γ

2 ∪ c

top

(

E

0,3,d

(3))

∩ [X

0,3,d

]

vir

. Denote by

φ

b

a

∈ H

2∗

_(X)

a lift of

φ

a

. ByDenition (2.25),wehave

e

_3∗

α =

s

′

₋₁

X

a=0

(e

_3∗

α, φ

a

)

red

φ

a

=

s

′

₋₁

X

a=0

e

_3∗

α, b

φ

a

tw

φ

a

Usingproje tion formula, the proposition follows from

hγ

1 , γ

2 , φ

a

i

red

0,3,d

=

hγ

1 , γ

2 ,

c

top

^

(

E)b

φ

a

i

0,3,d

=

e

_3∗

α, b

φ

a

tw

Dene the following onnexionon the bundle

F

:

∇

∂

_t0

:= ∂

t

0 +

1 z

1

• red

q

,

∀a ∈ {1, . . . , r}, ∇

δa

:= δ

a

+

1 z

T

a

• red

q

∇

δ

z

:= δ

z

−

1 z

E

• red

q

+µ

where

µ

isthe diagonalmorphismdened by

µ(φ

a

) :=

1

2 (deg(φ

a

)

− (dim

C

X

− rk E)) φ

a

and

E

_{:= t}

₀

1 _{+ c}

₁

₍

_T

_X

_{⊗ E}

∨

₎

.

Corollary 2.30. For any

γ

∈ H

2∗

_(X)

, we have :

(15)

Proof. This follows from Proposition 2.29 and from

µ(T

a

) = µ(T

a

)

. Lemma 2.31. For any

(t

0 , q, z)

in

H

0 _(X)

_{× V × C}

, we have :

L

tw

(t

0 , q, z)(ker m

c

top

) = ker m

c

top

.

Proof. Let

γ

be in

ker m

c

top

and

α

∈ H

2∗

_(X)

. Sin e

L

tw

_(t

0 , q, z)

is an automorphism of

H

2∗

_(X)

and

ker m

c

top

is the kernel of the twisted pairing

(

·, ·)

tw

we nd, using Proposition 2.20 :

α, L

tw

(t

0 , q, z)γ

tw

= L

tw

(t

0 , q,

−z).(L

tw

(t

0 , q,

−z))

−1

.α, L

tw

(t

0 , q, z)γ

tw

= L

tw

(t

0 , q,

−z)

−1

α, γ

tw

= 0.

Then

L

tw

_(t

0 , q, z)γ

belongs to

ker m

c

top

.

This lemma permit usto denea redu ed

L

fun tion : for any

(t

0 , q, z)

∈ V × C

put

L(t

0 , q, z) : H

2∗

(X)

−→ H

2∗

(X)

(2.32)

γ

7−→ L(t

0 , q, z)γ = L

tw

(t

0 , q, z)γ

Inthesamespiritof2.1.e,wealsogetanindu edintegralstru ture on

QDM(X,

E)

. Denote by

K(X) := K(X) /

_{{v | Ch(v) ∈ ker m}

c

top

}.

TheChern hara ter

Ch : K(X)

→ H

2∗

_(X)

indu esaredu edChern hara ter

Ch : K(X)

→

H

2∗

_(X)

whi h be ome anisomorphism aftertensored by

C

. Forany

v

∈ K(X)

, we put

Z(v) := (2π)

−(n−k)/2

_L(t

0 , q, z)z

−µ

z

c

1 (T

X

⊗E

∨

₎

b

Γ(

T

X

)b

Γ(

E)

−1

(2

√

−1π)

deg /2

_Ch(v).

Inthe same spiritofDenition2.21, the redu ed

b

Γ

-integralstru ture on

QDM(X,

E)

isgiven by

Z(K(X))

and we denote itby

F

Z

.

Corollary 2.33. The triple

(F ,

∇, S)

satises the following properties. 1. The onne tion

∇

isat and

S

is non-degenerated and

∇

-at.

2. A fundamental solution of

∇

is given by

L(t

0 , q, z)z

−µ

_z

c

1 (T

X

⊗E

∨

)

. 3. For any

s

1 , s

2 ∈ Γ(F)

, we have

S(L(q, z)s

1 , L(q, z)s

2 ) = S(s

1 , s

2 )

4. For any

v

in

K(X)

, we have

Z(v) = Z

tw

_(v).

5. For any

v

1 , v

2

in

K(X)

, we have

S(

_Z(v

1 ),

Z(v

2 )) =

Z

X

c

top

(

E) Td(T

X

) Td(

E)

−1

Ch(v

1 ⊗ v

2 ∨

).

Proof. (1)Proposition 2.17and Corollary2.30 impliesthe atnessfor

∇

. The atnessof

S

follows from Proposition2.20 and Equality (2.26).

(2) This statement follows easilyfromCorollary 2.30 and Proposition 2.17.

(3) The equality follows fromProposition 2.20 and Equality (2.26).

(4) This follows fromthe statement (2).

(5) Theequalityfollowsfromthe previousequality,Equation(2.26)andProposition2.23.