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Quantum Riemann-Roch, Lefschetz and SerreBy Tom Coates and Alexander Givental* To Vladimir Arnold on the occassion of his 70th birthday Abstract Given a holomorphic vector bundle E over

Annals of Mathematics

Quantum Riemann-Roch,

Lefschetz and Serre

By Tom Coates and Alexander Givental*

Quantum Riemann-Roch, Lefschetz and Serre

By Tom Coates and Alexander Givental*

To Vladimir Arnold on the occassion of his 70th birthday

Abstract

Given a holomorphic vector bundle E over a compact K¨ ahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers

in moduli spaces of stable maps f : Σ → X with the cap product of the virtual

fundamental class and a chosen multiplicative invertible characteristic class of

the virtual vector bundle H0(Σ, f ∗ E)  H1(Σ, f ∗ E) Using the formalism of

quantized quadratic Hamiltonians [25], we express the descendant potential

for the twisted theory in terms of that for X This result (Theorem 1) is

a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied tothe universal family over the moduli space of stable maps It determines alltwisted Gromov-Witten invariants, of all genera, in terms of untwisted invari-ants

When E is concave and the C×-equivariant inverse Euler class is chosen

as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E “Nonlinear Serre duality” [21], [23] expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE:

it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E ∗

We derive from Theorem 1 nonlinear Serre duality in a very general form(Corollary 2)

When the bundle E is convex and a submanifold Y ⊂ X is defined by

a global section of E, the genus-zero Gromov-Witten invariants of ΠE cide with those of Y We establish a “quantum Lefschetz hyperplane section

coin-principle” (Theorem 2) expressing genus-zero Gromov-Witten invariants of a

complete intersection Y in terms of those of X This extends earlier results

[4], [9], [18], [29], [33] and yields most of the known mirror formulas for toriccomplete intersections

*Research is partially supported by NSF Grants DMS-0072658 and DMS-0306316.

The mirror formula of Candelas et al [10] for the virtual numbers n d

of degree d = 1, 2, 3, holomorphic spheres on a quintic 3-fold Y ⊂ X =

CP4 can be stated [20] as the coincidence of the 2-dimensional cones over the

following two curves in Heven(Y ; Q) = Q[P ]/(P4):

JY (τ ) = e P τ +P

25

IY (t) =

d ≥0

e (P +d)t (5P + 1)(5P + 2) (5P + 5d)

(P + 1)5(P + 2)5 (P + d)5 .

The new proof given in this paper shares with earlier work [9], [18], [21],

[29], [33], [35] the formulation of sphere-counting in a hypersurface Y ⊂ X as

a problem in the Gromov-Witten theory of X.

Gromov-Witten invariants of a compact almost-K¨ahler manifold X are defined as intersection numbers in moduli spaces X g,n,d of stable pseudo-

holomorphic maps f : Σ → X Most results in this paper can be stated

and hold true in this generality (see Appendix 2 in [11]): the only exceptionsare those discussed in Sections 9 and 10 which depend on equation (19) Weprefer however to stay on the firmer ground of algebraic geometry, where themajority of applications belong

Given a holomorphic vector bundle E over a compact projective complex

manifold X and an invertible multiplicative characteristic class c of complex

vector bundles, we introduce twisted Gromov-Witten invariants as intersection indices in X g,n,d with the characteristic classes c(E g,n,d) of the virtual bundles

Eg,n,d = “H0(Σ, f ∗ E)H1(Σ, f ∗ E)” The “quantum Riemann-Roch theorem”

(Theorem 1) expresses twisted Gromov-Witten invariants (of any genus) andtheir gravitational descendants via untwisted ones

The totality of gravitational descendants in the genus-zero Gromov-Witten

theory of X can be encoded by a semi-infinite cone LX in the cohomology

alge-bra of X with coefficients in the field of Laurent series in 1/z (see §6) Another

such cone corresponds to each twisted theory LetLE be the cone ing to the total Chern class

correspond-c(·) = λdim(·) + c

1(·)λdim(·)−1 + + c

dim(·)(·).

Theorem 1 specialized to this case says that the cones LX andLE are related

by a linear transformation It is described in terms of the stationary phase

asymptotics a ρ (z) of the oscillating integral

1

√

2πz

 ∞0

e −x+(λ+ρ) ln x z dx

as multiplication in the cohomology algebra by 

i aρ i (z), where ρ i are the

Chern roots of E.

Assuming E to be a line bundle, we derive a “quantum hyperplane section

theorem” (Theorem 2) It is more general than the earlier versions [4], [29],

[18], [33] in the sense that the restrictions t ∈ H ≤2 (X;Q) on the space of

parameters and c1(E) ≤ c1(X) on the Fano index are removed.

In the quintic case when X = CP4 and ρ = 5P , the cone LX is known tocontain the curve

JX (t) =

d ≥0

e (P +zd)t/z (P + z)5 (P + zd)5,

and Theorem 2 implies that the coneLE contains the curve

IE (t) =

d ≥0

e (P +zd)t/z (λ + 5P + z) (λ + 5P + 5dz)

(P + z)5 (P + dz)5 .

One obtains the quintic mirror formula by passing to the limit λ = 0.

The idea of deriving mirror formulas by applying the Riemann-Roch theorem to universal stable maps is not new Apparently thiswas the initial plan of M Kontsevich back in 1993 In 2000, we had a chance

Grothendieck-to discuss a similar proposal with R Pandharipande We would like Grothendieck-to thankthese authors as well as A Barnard and A Knutson for helpful conversations,and the referee for many useful suggestions

The second author is grateful to D van Straten for the invitation to theworkshop “Algebraic aspects of mirror symmetry” held at Kaiserslautern inJune 2001 The discussions at the workshop and particularly the lectures on

“Variations of semi-infinite Hodge structures” by S Barannikov proved to bevery useful in our work on this project

1 Generating functions

Let X be a compact projective complex manifold of complex dimension D Denote by X g,n,d the moduli orbispace of genus-g, n-pointed stable maps [7], [31] to X of degree d, where d ∈ H2(X;Z) The moduli space is compact and

can be equipped [8], [34], [38] with a (rational-coefficient) virtual fundamental

cycle [Xg,n,d ] of complex dimension n + (1 − g)(D − 3) +d c1(T X).

The total descendant potential of X is a generating function for

Gromov-Witten invariants It is defined as

[X g,n,d](

Here ψ i is the first Chern class of the universal cotangent line bundle over X g,n,d

corresponding to the ith marked point, the map evi : X g,n,d → X is evaluation

at the ith marked point, t0, t1, ∈ H ∗ (X;Q) are cohomology classes, and

Q d is the representative of d ∈ H2(X;Z) in the semigroup ring of degrees of

formed by the operations of forgetting and evaluation at the last marked

point We pull E back to the universal family and then apply the K-theoretic push-forward to X g,n,d This means the following: there is a complex 0 →

E0

g,n,d → E1

g,n,d → 0 of bundles on Xg,n,d with cohomology sheaves equal to

R0π ∗(ev∗ n+1 E) and R1π ∗(ev∗ n+1 E) respectively Moreover, the difference

Eg,n,d := [E g,n,d0 ]− [E1

g,n,d]

in the Grothendieck group of bundles does not depend on the choice of the

complex These facts are based on some standard results about local complete

intersection morphisms, and are discussed further in Appendix 1.

A rational invertible multiplicative characteristic class of complex vectorbundles takes the form

where chk are components of the Chern character and s0, s1, s2, are

arbi-trary coefficients or indeterminates Given such a class and a holomorphic

vector bundle E ∈ K0(X) over X, we define the (c, E)-twisted descendant

potentials D c,E and F g

c,E by replacing the virtual fundamental cycles [X g,n,d]

in (1) and (2) with the cap-products c(E g,n,d)∩ [Xg,n,d] For example, thePoincar´e intersection pairing arises in Gromov-Witten theory as an intersec-

tion index in X 0,3,0 = X, and in the twisted theory therefore takes on the

1We will usually omit the prefix orbi.

We will often assume that all vector bundles carry the S1-action given

by fiberwise multiplication by the unitary scalars In this case the chk should

be understood as S1-equivariant characteristic classes, and all Gromov-Witten

invariants take values in the coefficient ring of S1-equivariant cohomology

the-ory We will always identify this ring H ∗ (BS1;Q) with Q[λ], where λ is the

first Chern class of the line bundle O(1) over CP ∞.

2 Quantization formalism

Theorem 1 below expresses D c,E in terms of DX via the formalism of

quantized quadratic Hamiltonians [25], which we now outline Consider H =

H ∗ (X;Q) as a super-space equipped with the nondegenerate symmetric linear form defined by the Poincar´e intersection pairing (a, b) = 

bi-X ab Let

H = H((z −1 )) denote the super-space of Laurent polynomials in 1/z with

coef-ficients in H, where the indeterminate z is regarded as even We equip H with

the even symplectic form

Ω(f , g) := 1

2πi (f (−z), g(z)) dz

=−(−1)¯f ¯ g

Ω(g, f ).

The polarization H = H+⊕ H − defined by the Lagrangian subspaces H+ =

H[z], H − = z −1 H[[z −1]] identifies (H, Ω) with the cotangent bundle T ∗ H+.The standard quantization convention associates to quadratic Hamiltoni-

ans G on ( H, Ω) differential operators ˆ G of order ≤ 2 acting on functions on

H+ More precisely, let {qα} be a Z2-graded coordinate system on H+ and

{pα} be the dual coordinate system on H −, so that the symplectic structure

in these coordinates assumes the Darboux form

pk(−z) −1−k is such a coordinate system In a Darboux

coor-dinate system the quantization convention reads

The quantization gives only a projective representation of the Lie algebra of

quadratic Hamiltonians on H as differential operators For quadratic

Hamil-tonians F and G we have

[ ˆF , ˆ G] = {F, G}ˆ + C(F, G),

where {·, ·} is the Poisson bracket, [·, ·] is the super-commutator, and C is the

cocycle

C(pαpβ , qαqβ) =

(−1)¯α¯β if α = β,

1 + (−1)¯α¯α if α = β,

C = 0 on any other pair of quadratic Darboux monomials.

We associate the quadratic Hamiltonian h T (f ) = Ω(T f , f )/2 to an infinitesimal

symplectic transformation T , and write ˆ T for the quantization ˆ hT If A and B

are self-adjoint operators on H then the operators f

on H are infinitesimal symplectic transformations, and

con-defined Such special circumstances usually involve -adic convergence with respect to some auxiliary formal parameters (such as s k in Corollary 3, 1/λ in (12), Q in (13), etc.) The key point here is that our formulas provide unam-

biguous rules for transforming generating functions (and their coefficients): thedescription of these rules as symplectic transformations or their quantizationsremains merely a convenient interpretation2

Let us begin by setting up notation for such an interpretation We willassume that the ground field Q of constants is extended to the Novikov ring

Q[[Q]], or to Q[[Q]] ⊗ Q(λ) in the S1-equivariant setting, and will denote theground ring by Λ The potentialsF g

X (t0, t1, ) are naturally defined as formal

functions on the space of vector polynomials t(z) = t0 + t1z + t2z2 + where t0, t1, t2, ∈ H The total descendant potential DX is simply the

formal expression exp

g −1 F g

X defined by these formal functions It cannot

be viewed as a formal function of
and t because of the presence of
−1

-and
0-terms in the exponent The reader uncomfortable with this situationcould note that the formal functions F0

X when reduced modulo Q

contain only terms which are respectively at-least-cubic and at-least-linear in

2 This approach, somewhat resembling the terminology in the theory of formal groups, is

not the only one possible We refer to Section 8 in [26] where the class of tame asymptotic

functions (convenient for the purposes of that paper) is introduced.

the variables t i, and thatDX can therefore be considered as a formal function

of
, t/
and Q/
This point of view will, however, play no role in what

the super-space H equipped with the twisted inner products (4) Alternatively,

we can identify the inner product spaces (H, ( ·, ·) c(E) ) with (H, ( ·, ·)) by means

of the maps a 

c(E), hence considering the twisted descendant potentials

D c,E as asymptotic elements of the original Fock space via the twisted dilaton

shift:

q(z) =

c(E) (t(z) − z).

(6)

We thus obtain a formal familyDs:=D c,E of asymptotic elements of the Fock

space depending on the parameters s = (s0, s1, s2, ) from (3) Note that,

due to the dilaton shift, Ds is a formal function of q defined near the shifted

origin q(z) = −c(E)z, which varies with s.

.

The operator of multiplication by chl (E) in the cohomology algebra H

of X is self-adjoint with respect to the Poincar´e pairing Consequently, the

operator of multiplication by chl (E)z 2m −1 in the algebra H is an infinitesimal

symplectic transformation of H and so is ln  Theorem 1 therefore is derived

from the following more precise version

c(E)

−1 24

Here sdet(·) = exp str ln(·) is the Berezinian.

Remarks (1) The variable s0 is present on the RHS of (7) only in the

form exp(s0ρ/z)ˆ where ρ = ch1(E) For any ρ ∈ H2(X) the operator (ρ/z)ˆ

is in fact a divisor operator: the total descendant potential satisfies the divisor

Here Q i are generators in the Novikov ring corresponding to a choice of a

basis in H2(X) and ρ i are coordinates of ρ with respect to the dual basis For ρ = ch1(E) the c D −1 -term cancels with the s0-term on the LHS of (7)

Thus the action of the s0-flow reduces to the change Q d d exp(s0



d ρ) in

the descendant potential DX combined with the multiplication by the factor

exp (s0(dim E)χ(X)/48) coming from the super-determinant.

(2) If E = C then E g,n,d =C − E ∗

g, where Eg is the Hodge bundle The

Hodge bundles satisfy chk(Eg) = − chk(E∗ g) In view of this, Theorem 1 in

this case turns into Theorem 4.1 in [25] and is a reformulation in terms of the

formalism explained in Section 2 of the results of Mumford [36] and Pandharipande [16] The proof of Theorem 1 is based on a similar applica-tion of Mumford’s Grothendieck-Riemann-Roch argument to our somewhatmore general situation The argument was doubtless known to the authors of[16] The main new observation here is that the combinatorics of the result-ing formula, which appears at first sight rather complicated, fits nicely withthe formalism of quantized quadratic Hamiltonians A verification of this —somewhat tedious but straightforward — is presented in Appendix 1

Faber-4 The Euler class

The S1-equivariant Euler class of E is written in terms of the ariant) Chern roots ρ i as

i

exp

124

ρ ln λ +
(−1) k −1 ρ k+1

k(k + 1)λ k =

 ρ0

ln(λ + x)dx = [(λ + x) ln(λ + x) − (λ + x)] | ρ

0.

It has positive cohomological degree and is small in this sense The constant

term (λ ln λ − λ)/z is thrown away on the following grounds According to

[25], (1/z)ˆ is the string operator and annihilates the descendant potential DX

Thus the operators exp((λ ln λ − λ)/z)ˆ) do not change DX The rest of the

series in the exponent converges in the 1/λ-adic topology.

Corollary 2 We have D ∗

s = (sdet c(E)) −1/24 Ds More explicitly,

Dc∗ ,E ∗(t∗ ) = (sdet c(E)) −241 D c,E (t),

where

t∗ (z) = c(E)t(z) + (1 − c(E))z.

Proof Replacing chl (E) with ( −1) lchl (E), and s k with (−1) k+1 sk in (7)preserves all terms except the super-determinant

Corollary 3 Consider the dual bundle E ∗ equipped with the dual

S1-action, and the S1-equivariant inverse Euler class e −1 Put

t∗ (z) = z + ( −1) dim E/2 e(E)(t(z) − z) and introduce the change ± : Q d d(−1)dch1(E) in the Novikov ring With this notation

k = (−1) k+1 sk as in the situation of Corollary

2 However, s ∗0 =−s0− π √ −1 We compensate for the discrepancy −π √ −1

using the divisor equation (8) described in Remark 1 following Theorem 1

6 The genus-zero picture

The genus-zero descendant potential F0

X can be recovered from the called “J-function” of finitely many variables due to a reconstruction theoremessentially due to Dubrovin [14] and going back to Dijkgraaf and Witten [13]

so-The J-function is a formal function of t ∈ H and 1/z with vector coefficients

in H defined by

∀a ∈ H, (JX (t, z), a) := (z + t, a) +

d,n

Q d n!

z − ψn+1 .

(11)

We need the following reformulation of the reconstruction theorem in terms ofthe geometry of the symplectic space (H, Ω).

The genus-zero descendant potential F0

X considered as a formal function

of q∈ H+via the dilaton shift (5) generates (the germ of) a Lagrangian section

mov-(i) tangent spaces L ⊂ H to LX are tangent to LX along zL and, vice versa,

if L = TfL is a tangent space to L then f is contained in zL;

(ii) J X (t, −z) ∈ H is the intersection of LX with (t − z) + H−

Remarks (1) LX ⊂ T ∗ H+is a formal germ of a Lagrangian section defined

near q =−z All geometric statements about LX should be understood in thesense of formal geometry

(2) Part (i) of the proposition implies that the tangent spaces L are Lagrangian subspaces satisfying zL ⊂ L (as well as zL ⊂ LX) They con-

sequently belong to the loop group Grassmannian of the “twisted” series A(2),

or to its super-version

(3) Part (i) of the proposition means that the tangent spaces L actually depend only on dim H parameters and form a semi-infinite variation of Hodge

structure in the sense of [3].

(4) Part (i) follows easily from Dubrovin’s reconstruction formula (see

[14, Th 6.1]) in the axiomatic theory of Frobenius structures We refer to [27]

for details In Appendix 2 we give another, more direct proof applicable in

Gromov-Witten theory It is based on Theorem 5.1 stated in [25] which relates gravitational descendants with ancestors.

(5) Part (ii) of the proposition follows immediately from the definitions of

JX and LX Together with part (i) it shows how to reconstruct the cone LX

from the J-function Namely, the first t-derivatives of J X (t, z) form a basis in the intersection of the tangent space L to LX with z H− and therefore form a

basis of L as a free Λ[z]-module We describe this reconstruction procedure in

more detail in Section 8

In the quasi-classical limit
→ 0, quantized symplectic transformations

exp ˆA of Theorem 1  acting on the total potentialsDsconsidered as asymptoticelements of the Fock space turn into “unquantized” symplectic transformationsacting byLs son the Lagrangian conesLsgenerated by the genus-zero potentialsF0

c,E.Corollary 4

In the case of genus-zero Gromov-Witten theory twisted by the Euler

class e(E), the corresponding Lagrangian cone Le is obtained from LX bymultiplication inH by the product over the Chern roots ρ of the series

The series (12) is well-known [28] in connection with the asymptotic expansion

of the gamma function Γ((λ + ρ)/z) More precisely, (12) coincides with the

stationary phase asymptotics of the integral

1



2πz(λ + ρ)

 ∞0

e −x+(λ+ρ) ln x z dx

near the critical point x = λ + ρ of the phase function.

Let us assume now that E is the direct sum of r line bundles with first Chern classes ρ i — in what follows we will need the Chern roots to be definedover Z — and consider the J-function J X (t, z) =

Theorem 2 The hypergeometric modification IE , considered as a

fam-ily t E (t, −z) of vectors in the symplectic space (H, Ω e(E) ) corresponding

to the twisted inner product (a, b) e(E) = 

X e(E)ab on H, is situated on the

Lagrangian section L e,E ⊂ (H, Ω e(E)) defined by the differential of the twisted

genus-zero descendant potential F0

e,E

Note that in defining L e,E we regard F0

e,E as a formal function of q via

the (untwisted) dilaton shift Also, the following comment is in order The

se-ries I E does not necessarily belong to H((z −1)) because of possible unbounded

growth of the numbers ρ i (d) However the coefficients of each particular mial Q ddo Similarly, multiplication by the series (12) moves the coneLX out

mono-of the space H((z −1 )) However modulo each particular power of 1/λ it does not (the invariance of the cone with respect to the string flow exp(λ ln λ −λ)/z

is once again essential here) In fact all our formulas make sense as ations with generating functions (i.e give rise to legitimate operations with

oper-their coefficients) because of the presence of suitable auxiliary variables — s k

in Corollary 3, 1/λ in (12), Q in (13) More formally, this means the following.

We replace the ground ring Λ in H = H ∗ (X, Λ) with its completion (which we

will still denote Λ) in the appropriate (s-adic, 1/λ-adic, Q-adic) topology In

the role of the symplectic space H, we should take the space (we will denote

it H {{z −1 }}) of Laurent series k ∈Z hkz k possibly infinite in both directions

but satisfying the following convergence condition: as k → +∞, hk → 0 in the

topology of Λ In the following proof we will have to similarly replace Λ[z] by

Λ{z}, and the ring Λ should also be extended by √ λ.

8 Proof of Theorem 2

Due to the equivariance properties (see [21,§6]) of J-functions with respect

to the string and divisor flows (8) we have

dx1 .

 ∞0

e(E) identifies the Lagrangian cone L e,E ⊂

(H, Ω e(E)) with its normalized incarnation Le⊂ (H, Ω) Therefore Theorem 2

is equivalent to the inclusion

IE (t, −z)e(E) ∈ Le

and hence

IE (t, −z)e(E)

bρ i(−z) ∈ LX

due to Corollary 4 It remains to show therefore that the asymptotic expansion

of the integral (15) belongs to the cone determined by the J-function J X (t, z).

In fact we will prove the following

Lemma For each t, the asymptotic expansion of the integral (15) differs

from λ dim E/2 JX (t ∗ , z) (at some other point t ∗ (t)) by a linear combination of

the first t-derivatives of JX at t ∗ with coefficients in zΛ{z}.

For this, we are going to use another property of the J-function J X known in quantum cohomology theory and in the theory of Frobenius struc-tures (see for instance [21, §6] and [14]) The first derivatives ∂JX /∂t α satisfythe system of linear PDEs

where we use a coordinate system t = 

t α φα on H Indeed, we can argue

as in [3] The first t-derivatives of J X form a basis in the intersection of the

tangent space L to the cone LX with z H − The LHS of (16) belongs to this

intersection: it is in L since infinitesimal t-variations of zL are in L, and it is

in z H− since J X ∈ z + t + H−

Further analysis reveals that A γ αβ are structure constants of the quantum

cohomology algebra φ α • φβ = 

A γ αβ φα In particular, z∂1JX = J X since

1• = id We use the notation ∂v for the directional derivative in the direction

of v ∈ H and take here v = 1.

We can interpret (16) as the relations defining the D-module generated by

JX, i.e obtained from it by application of all differential operators Using

Tay-lor’s formula J X (t + yρ) = exp(y∂ ρ )J X (t) we now view (15) as the asymptotic

expansion of the oscillating integral taking values in this D-module:

(17) (2πz) − r2

 ∞0

dx1 .

 ∞0

Using this (and also the relation λJ X = z∂ λ ·1 JX mentioned earlier) we see that

(λ + ρ i•) ln(λ + ρi•) − (λ + ρi•) ] 1 Processing next

the factor e12ln(λ+z∂ ρi), we take out √

λ The remaining factor e12ln(1+z∂ ρi /λ)

together with the rest of the exponent in the asymptotic expansion (17) yields

an expression of the type e o(z)/z JX (t ∗ , z) too We conclude that the expansion

(17) assumes the form

λ dim E2 JX (t ∗ , z) +

α

Cα (t ∗ , z) z∂φ α JX (t ∗ , z),

where the coefficients C α (t ∗ , ·) are in Λ{z} as required.

Remark. The proof of the lemma actually shows that given a family

Φ(x, p) of phase functions parametrized by p ∈ H ∗ the asymptotic expansion

of the oscillating integral

dx e Φ(x,z∂)/z JX (t, z) belongs to the same cone as J X

Thus we have proved that the vector I E (t, −z) is situated on the

Lagrangian cone L e,E It therefore differs from the value of the

correspond-ing J-function Je,E(τ, −z) at a suitable point τ = τ(t) by a linear

combina-tion of the derivatives ∂Je,E /∂τ α with coefficients in zΛ {z} Moreover, these

derivatives form a basis in the tangent space L to L e,E considered as a free

Λ{z}-module, and so the derivatives ∂IE /∂t α ∈ L are expressible as their

linear combinations The last statement is equivalent to the Birkhoff

factor-ization U (z, z −1 ) = V (z −1 )W (z) where the columns of the matrix U are the derivatives of I E , and those of V are the derivatives of Je,E.

Let us use now the obvious fact that modulo the Novikov variables Q the functions I E and Je,E coincide (at t = τ ) and hence W (z) turns into the identity matrix in this specialization Thus det W ∈ 1 + Q Λ{z} is invertible

in Λ{z} and therefore we can write V = UW −1 Together with the expression

J e,E = z∂1J e,E of the function z −1 J e,E as one of the columns of the matrix V

this proves existence of the representation (18) in the following corollary.Corollary 5 Let L e,E ⊂ (H, Ω e(E)) be the Lagrangian cone determined

by the J-function J e,E corresponding to (e, E)-twisted Gromov-Witten theory,

and let Lt be the tangent space to L e,E at the point IE (t, −z) Then the tersection (unique due to some transversality property) of zLt with the affine subspace −z + zH − coincides with the value J e,E(τ, −z) ∈ −z + τ(t) + H − of the J-function In other words,

in-J e,E(τ, z) = I E (t, z) +

α

cα (t, z) z ∂ φ α IE (t, z), where c α (t, ·) ∈ Λ{z},

(18)

and τ (t) is determined as the z0-mode of the RHS.

Remark. A by-product of Corollary 5 is a geometrical description of

(−z + zH − ) comes naturally parametrized by t which may have little in mon with the projections τ − z of the intersection points along H −

com-9 Mirror formulas

Let us assume now that the bundle E (which is still the sum of line bundles with first Chern classes ρ i ) is convex, i.e spanned fiberwise by global

sections, and apply the above results to the genus-zero Gromov-Witten theory

of a complete intersection j : Y → X defined by a global section While the

above proof of Theorem 2 fails miserably in the limit λ = 0, the definition of the series Je,E and I E and the relation between them described by Corollary

5 survive the nonequivariant specialization Namely, at λ = 0 the J-function

J e,E degenerates into

JX,Y (t, z) = z + t +

d,n

Q d n! (evn+1)

where (evn+1) is the cohomological push-forward along the evaluation map

evn+1 : X 0,n+1,d → X and e is the (nonequivariant!) Euler class Here

E 0,n+1,d  ⊂ E 0,n+1,d is the subbundle defined as the kernel of the evaluation

map E 0,n+1,d → ev ∗

n+1 E of sections (from H0(Σ, f ∗ E)) at the (n+1)st marked

point

The function J X,Y is related to the Gromov-Witten invariants of Y by

e(E)J X,Y (u, z) = H2(Y )→H2(X)j ∗ JY (j ∗ u, z),

(19)

since [Y 0,n+1,d ] = e(E 0,n+1,d) ∩ [X 0,n+1,d] (see for instance [30]) The longsubscript here is to remind us that the corresponding homomorphism betweenNovikov rings should be applied to the RHS

On the other hand, the series I E in the limit λ = 0 specializes to

since ρ i (d) ≥ 0 for all degrees d of holomorphic curves Passing to the limit

λ = 0 in Theorem 2 and Corollary 5 we obtain the following “mirror theorem”.

Corollary 6 The series IX,Y (t, −z) and JX,Y (τ, −z) determine the same cone In particular, the series JX,Y related to the J-function of Y by (19) is recovered from IX,Y via the Birkhoff factorization procedure followed by the mirror map t

Remark Corollary 6 is more general than the (otherwise similar)

quan-tum Lefschetz hyperplane section theorems by Bertram and Lee [9], [33] andGathmann [18] for

(i) it is applicable to arbitrary complete intersections Y without the tion c1(Y ) ≥ 0, and

restric-(ii) it describes the J-functions not only over the small space of parameters

t ∈ H ≤2 (X, Λ) but over the entire Frobenius manifold H ∗ (X, Λ).

In fact the results of [18] allow one to deal with both generalizations and

to compute recursively the corresponding Gromov-Witten invariants one at atime What has been missing so far is the part that Birkhoff factorization plays

in the formulations

Now restricting J X,Y and I X,Y to the small parameter space H ≤2 (X, Λ) and assuming that c1(E) ≤ c1(X) we can derive the quantum Lefschetz theo-

rems of [4], [9], [18], [29], [33] A dimensional argument shows that the series

IX,Y on the small parameter space has the form

IX,Y (t, z) = zF (t) +

G i (t)φ i + O(z −1 ),

where{φi} is a basis in H ≤2 (X, Λ), G i and F are scalar formal functions, and

F is invertible (we have F = 1 and G i = t i when the Fano index is not toosmall)

Corollary 7 When c1(E) ≤ c1(X) the restriction of J X,Y to the small parameter space τ ∈ H2(X, Λ) is given by

JX,Y (τ, z) = IX,Y (t, z)

F (t) , where τ =

G i (t)

F (t) φi.

The J-function of X = CP n −1 restricted to the small parameter plane

t0+ tP (where P is the hyperplane class generating the algebra H ∗ (X, Λ) = Λ[P ]/(P n)) takes the form

expan-Projecting J X,Y by j ∗ onto the cohomology algebra Λ[P ]/(P n −1) ⊂

H ∗ (Y, Λ) we recover the mirror theorem of [25] and, in the case l = n = 5, the quintic mirror formula of Candelas et al [10].

10 Serre duality in genus zero

LetL c,E be the Lagrangian cone in the symplectic space (H, Ω c(E)) defined

by the genus-zero descendant potential F0

c,E, and Lc∗ ,E ∗ be the Lagrangiancone in the symplectic space (H, Ωc∗ (E ∗)) defined by the genus-zero descendant

of linear symplectic spaces identifies Lc∗ ,E ∗ with L c,E

In particular, the family

Proof To simplify the notation put J := J c,E, J ∗ := Jc ∗ ,E ∗ , c := c(E),

c ∗ := c∗ (E ∗ ) = c −1 There exist coefficients C α (which could a priori be polynomial in z and depend on τ ∗ but turn out here to be constant) and a

change of variables τ = τ (τ ∗), such that

We can repeat the above arguments in the situation of Corollary 3 where

c = e is the S1-equivariant Euler class

Corollary 11 The map f dim Ee−1 (E)f , Q

Le−1 ,E ∗ with L e,E Furthermore,

e(E)Je,E(τ, z; Q) = z( −1) dim E ∂ e(E) Je−1 ,E ∗ (τ ∗ , z; ±Q),

where for all φ ∈ H we have (τ, φ) = ∂φ∂ e(E) F0

e−1 ,E ∗ (τ ∗ , 0, 0, ; ±Q).