Quantum cohomology of blowups of surfaces and its functoriality property

Jianxun Hu^{1}

Department of Mathematics, Zhongshan University, Guangzhou, 510275 P. R. China and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Abstract

In this paper, using the WDVV equation, one first proved that all Gromov-Witten invariants of blowups of surfaces can be computed from the Gromov-Witten invariants of itself by some recursive relations. Furthermore, one may determine the quantum product on blowups. One also proved that there is some degree of functoriality of the big quantum cohomology for a blowup.

1 Introduction

Historically, the idea of quantum cohomology first appeared in the physical literature in the guise of the chiral ring of a Calabi-Yau 3-fold. The chiral ring was understood as a deformation of the ordinary cohomology ring [LVW], agreeing with the ordinary cohomology ring at large radius limit points. The quantum product can be understood more generally in the context of topological quantum field theories [W1, W2]. The mathematical foundation of quantum cohomology was first established for semi-positive symplectic manifold by Ruan and Tian [RT]. There actually are two types of quantum cohomology: the small and big quantum cohomology. In this paper, we will only consider the later.

One can use Gromov-Witten(GW) invariants to define the big quantum cohomology ring QH^{∗}(X) of a compact symplectic manifold X. It is wellknown that the quantum product is associative. Moreover, the associativity is equivalent to a certain partial differential equation satisfied by the generating function Φ, the so-called WDVV equation [W2, DVV1, DVV2]. WDVV equation gives rise to some relations among the GW invariants which often are sufficient to recursively determine all GW invariants from a few basic ones. This approach was used to determine the numbers N_{d} of nodal rational curves of degree d in projective plane in [KM] and [RT]. In section 3, we will first use the WDVV equation to obtain some recursive relation among GW invariants of blowups of nonsingular algebraic surfaces. Therefore, we may compute all GW invariants of the blowup of X from ones of X. Furthermore, we may determine the quantum product on the blowups. Quantum cohomology of blowups of surfaces was considered by Crauder-Miranda [CM] and Göttsche-Pandharipande[GP]. The method of [CM] is fully algebro-geometric. Our method is the same as that of [GP]. They only consider blowups of P^{2}, but we consider general surfaces. We actually generalize the reaults of [GP] to blowups of general surfaces.

1supported by Zhongshan university advanced research center foundation

22000 AMS Classification: Primary 53D45. Secondary 14N35

So far, we know little about the naturality of quantum cohomology [R1, R2]. To understand its naturality property, we also need to understand the change of GW invariant under some symplectic morphisms. It is wellknown that blowup is a type of fundamental surgeries in the category of algebraic and symplectic geometry. In section 4, we will show that there is some degree ( i. e. module the effect of homology ) of functoriality property of the big quantum cohomology for a blowup.

Acknowledgement: The author would like to thank Prof. Yongbin Ruan for his many suggestive discussion and encouragement.

2 Quantum cohomology and WDVV equation

Let X be a smooth projective manifold of dimension 2n, ω be a complexified Kahler class on X, β ∈ H_{2}(X) and k be a positive integer. Denote by \overline{M}_{k}(X, β) the moduli space of genus zero k-pointed stable maps f : C ⟶ X with the homology class f_{∗}[C] = β. It is wellknown that \overline{M}_{k}(X, β) has a virtual fundamental class, denoted by [\overline{M}_{k}(X, β)]^{V ir}, with the expected dimension.

There are k evaluation maps

ev_{i} : \overline{M}_{k}(X, β) ⟶ X

for each marked point x_{1}, ⋯, x_{k}. Given cohomology classes γ_{1}, ⋯, γ_{k} ∈ H^{∗}(X), we can define the Gromov-Witten invariant as

Ψ_{β}(γ_{1}, ⋯, γ_{k}) = ∫_{[\overline{M}_{k}(X,β)]^{V ir}} Π_{i}ev_{i}^{∗}γ_{i}.

We fix a basis T_{0} = 1, T_{1}, ⋯, T_{m} of the rational cohomology H^{∗}(X, Q). Let g_{ij} = ∫_{X} T_{i} ∪ T_{j} and then (g^{ij}) = (g_{ij})^{−1} is the inverse matrix.

Put γ = ∑_{i=0}^{m} t_{i}T_{i} where t_{i} is a supercommutative variable with deg t_{i} = deg T_{i}. Here supercommutativity means that

t_{i}t_{j} = (−1)^{deg t_{i} deg t_{j}}t_{j}t_{i}

and t_{i}^{2} = 0 if t_{i} has odd degree.

We define the Gromov-Witten potential Φ as the formal series

Φ(t_{0}, t_{1}, ⋯, t_{m}) := ∑_{n≥3} ∑_{β∈H_{2}(X,Z)} \frac{1}{n!}Ψ_{β}(γ^{n})q^{β}
= \frac{1}{6} ∫_{X} γ^{3} + ∑_{n≥3} ∑_{β∈H_{2}(X,Z)−\{0\}} \frac{1}{n!}Ψ_{β}(γ^{n})q^{β}

where q^{β} = e^{2πi ∫_{β} ω}.

To define the quantum product in terms of GW potential Φ, we will need to avoid the convergence problem of some formal series like Φ. We will regard q^{β} as a formal variable rather than a number. More precisely, by the effectivity axiom, we can assume that β lies in the integral Mori cone M(X)_{Z}. Then we consider

R = Q[[q^{β}; β ∈ M(X)_{Z}]],

where the double brackets indicate that we allow infinite sums. The product of two such elements is

∑_{M(X)_{Z}} a_{β}q^{β} · ∑_{M(X)_{Z}} b_{β}q^{β} = ∑_{M(X)_{Z}}( ∑_{β=β_{1}+β_{2}} a_{β_{1}}b_{β_{2}})q^{β}.

This sum makes sense because for any β ∈ M(X)_{Z}, the sum is parentheses is finite since the free part of the Mori cone consists of lattice points in a strongly convex cone. Thus R is a ring.

Let T^{k} = ∑_{l} g^{kl}T_{l}. Then we define the quantum product of T_{i} and T_{j} as follows:

T_{i} ∗ T_{j} = ∑_{k} \frac{∂^{3}Φ}{∂t_{i}∂t_{j}∂t_{k}} T^{k} = ∑_{k,l} \frac{∂^{3}Φ}{∂t_{i}∂t_{j}∂t_{k}} g^{kl}T_{l}.

Extending this linearly gives the big quantum product on the cohomology H^{∗}(X, R). Indeed, H^{∗}(X, Λ(X, R) is a ring under the big quantum product. Moreover, the quantum product is associative. We call H^{∗}(X, R) with the quantum product as the big quantum cohomology of X.

Kontsevich-Manin [KM] and Ruan-Tian [RT] proved that the associative property of big quantum cohomology is equivalent to a certain partial differential equation satisfied by Φ, the so-called WDVV equation. That means that the Gromov-Witten potential Φ satisfies the WDVV equation

∑_{a,b} \frac{∂^{3}Φ}{∂t_{i}∂t_{j}∂t_{a}} g^{ab} \frac{∂^{3}Φ}{∂t_{b}∂t_{k}∂t_{l}}

= (−1)^{deg t_{i}(deg t_{j}+deg t_{k})} ∑_{a,b} \frac{∂^{3}Φ}{∂t_{j}∂t_{k}∂t_{a}} g^{ab} \frac{∂^{3}Φ}{∂t_{b}∂t_{i}∂t_{l}}

for all i, j, k, l. This WDVV equation will give rise to some recursive relations among GW invariants.

3 Quantum cohomology of blowups of surfaces

In the following sections, we will assume X is a nonsingular semipositive projective surface with the Kahler form as the symplectic structure ω. For simplicity we also assume that the Chow and homology rings of X coincide. Let X_{r} be the blowup of X in r generic points x_{1}, ⋯, x_{r}. Let m = r + p + 1. Let T_{0} = 1, T_{1}, ⋯, T_{p}, T_{i+p}(i = 1, ⋯, r), T_{m} be a basis of H^{∗}(X_{r}, Q), where T_{1}, ⋯, T_{p} is a basis of H^{2}(X). T_{i+p}, (i = 1, ⋯, r) are respectively the Poincare dual cohomology classes of the exceptional divisors E_{i}. T_{m} is the class of a point.

For an r-tuple α = (a_{1}, ⋯, a_{r}) of integers, denote by (β, α) the homology class β − ∑_{i=1}^{r} a_{i}E_{i}, where β is the pullback of a homology class of X. In the following, we will use the same notation to denote the homology class and its pullback. Let |α| = ∑_{i=1}^{r} a_{i}. Let n_{β,α} = C_{1}(β) − |α| − 1 be the expected dimension of the moduli space \overline{M}_{0}(X_{r}, (β, α)), where C_{1} is the first Chern class of X. If n_{β,α} ≥ 0, let

N_{β,α} = Ψ_{(β,α)}(T_{m}^{n_{β,α}})

be the corresponding GW invariant. When writing N_{β,α} for a sequence α of length r, we will always mean the GW invariant on X_{r}. For any r, we write [i]_{r} = (j_{1}, ⋯, j_{r}) with j_{k} = δ_{ik}. For any α = (a_{1}, ⋯, a_{r−1}), we denote by (α, k) the sequence (a_{1}, ⋯, a_{r−1}, k). For a permutation σ of {1, ⋯, r}, denote by α_{σ} the sequence (a_{σ(1)}, ⋯, a_{σ(r)}). For an integer k, we write α ≥ k to mean that a_{i} ≥ k for all i.

About the change of GW invariants under blowup, the author[H1] proved the following lemma.

Lemma 3.1: Let X be a compact symplectic manifold of dimension 2n. Suppose β ∈ H_{2}(X), γ_{1}, ⋯, γ_{k} ∈ H^{∗}(X). Denote by p : X̃ ⟶ X the projection of the blowup, Then we have

(i) Ψ_{β}^{X}(γ_{1}, ⋯, γ_{k}) = ΨX̃ ; β(p^{∗}γ_{1}, …, p^{∗}γ_{k}).

(ii) Ψ_{β}^{X}(γ_{1}, ⋯, γ_{k}, [pt]) = ΨX̃ ; β−e(p^{∗}γ_{1}, ⋯, p^{∗}γ_{k}).

where e the homology class of a line in the exceptional divisor.

¿From Lemma 3.1, we have N_{β,α} satisfy the following properties:

(1) N_{β,α} = N_{β,(α,0)}

(2) If n_{β,α} > 0, then N_{β,α} = N_{β,(α,1)}.

It is wellknown that All effective classes (β, α) on X_{r} satisfy α ≤ C_{1}(β).We hope to obtain some recursion formulae determining the N_{β,α} from the associativity of the quantum product.

Let γ = t_{0}T_{0} + δ + ϵ where δ = ∑_{i=1}^{r+p} t_{i}T_{i} and ϵ = t_{m}T_{m}. Then the GW potential of X_{r} is

Φ = \frac{1}{6} ∫_{X_{r}} γ^{3} + ∑_{n≥3} ∑_{(β,α)≠0} \frac{1}{n!}Ψ_{(β,α)}((δ + ϵ)^{n})q^{(β,α)}

= \frac{1}{6} ∫_{X_{r}} γ^{3} + ∑_{(β,α)≠0} N_{β,α}e^{∑_{i=1}^{p} t_{i} ∫_{β} T_{i}}e^{∑_{i=1}^{r} a_{i}t_{i+p}} \frac{t_{m}^{n_{β,α}}}{n_{β,α}!}q^{β}q_{1}^{a_{1}} ⋯ q_{r}^{a_{r}}

where the summation runs over all (β, α) ≠ 0 satisfying n_{β,α} ≥ 0, C_{1}(β) ≥ 0 and α ≤ C_{1}(β).

Let Φ_{ijk} = ∂_{i}∂_{j}∂_{k}Φ where ∂_{i} = \frac{∂}{∂t_{i}}. Then the quantum product of T_{i} and T_{j} is given by

T_{i} ∗ T_{j} = (T_{i} · T_{j})T_{m} + ∑_{k,l=1}^{p} Φ_{ijl}g^{lk}T_{k} − ∑_{k=p+1}^{p+r} Φ_{ijk}T_{k} + Φ_{ijm}T_{0}.  (3.1)

Therefore, from the WDVV equation, it is easy to prove the following lemma.

Lemma 3.2: For i, j, k, l ∈ {1, ⋯, m}, there is a relation:

(R_{i,j,k,l})  (T_{i} · T_{j})Φ_{mkl} + (T_{k} · T_{l})Φ_{ijm} − (T_{j} · T_{k})Φ_{mil} − (T_{i} · T_{l})Φ_{jkm}
= ∑_{a,b=1}^{p}(Φ_{jka}g^{ab}Φ_{bil} − Φ_{ija}g^{ab}Φ_{bkl}) + ∑_{a=p+1}^{p+r}(Φ_{ija}Φ_{akl} − Φ_{jka}Φ_{ail}).

For the purpose of the recursive determination of the N_{β,α}, we first obtained that for i, j ∈ {1, ⋯, p}

(R_{i,j,m,m})  (T_{i} · T_{j})Φ_{mmm} =

∑_{a,b=1}^{p}(Φ_{jma}g^{ab}Φ_{bim} − Φ_{ija}g^{ab}Φ_{bmm}) + ∑_{s=p+1}^{p+r}(Φ_{ijs}Φ_{smm} − Φ_{jms}Φ_{sim})

and for all k = p + 1, ⋯, p + r, i, j ∈ {1, ⋯, p},

(R_{i,j,k,k})  (T_{i} · T_{j})Φ_{mkk} − Φ_{ijm} =

∑_{a,b=1}^{p}(Φ_{jka}g^{ab}Φ_{bik} − Φ_{ija}g^{ab}Φ_{bkk}) + ∑_{a=p+1}^{p+r}(Φ_{ija}Φ_{akk} − Φ_{jka}Φ_{aik}).

Remark: It is easy to see that in case X = P^{2}, the relation R_{i,j,k,l} coincides with R_{i,j,k,l} of [GP].

Let the symbol ⊢ (β, α) denote the set of pairs ((β_{1}, α_{1}), (β_{2}, α_{2})) satisfying

(i) (β_{1}, α_{1}), (β_{2}, α_{2}) ≠ 0. (ii) (β_{1}, α_{1}) + (β_{2}, α_{2}) = (β, α). (iii) n_{β_{1},α_{1}}, n_{β_{2},α_{2}} ≥ 0, C_{1}(β_{1}), C_{1}(β_{2}) ≥ 0, α_{1} ≤ C_{1}(β_{1}), α_{2} ≤ C_{1}(β_{2}).

The notation ⊢ (β, α) ≠ 0 will be used to denote the subset of ⊢ (β, α) satisfying β_{1} ≠ 0, β_{2} ≠ 0. The binomial coefficient ( \begin{matrix}p \\ q \end{matrix} ) is defined to be zero if q < 0 or p < q.

Theorem 3.3: The N_{β,α} are determined by the initial values

(i) N_{(β,0)} = Ψ_{β}^{X}(T_{m}^{n_{β}}) for all r. (ii) N_{0,−[i]_{r}} = 1, for i ∈ {1, ⋯, r}. and the following recursion relations.

If n_{β,α} ≥ 3, choosing i, j ∈ {1, ⋯, p} such that g_{ij} ≠ 0, then relation R(m) holds:

(R(m))  N_{β,α} =

\frac{1}{g_{ij}} ∑_{⊢(β,α)≠0} N_{β_{1},α_{1}}N_{β_{2},α_{2}} ( ∑_{a,b=1}^{p} T_{a}(β_{1})g^{ab}T_{b}(β_{2}) − ∑_{s=1}^{r} b_{s}c_{s})

[T_{i}(β_{1})T_{j}(β_{2}) ( \begin{matrix}n_{β,α} − 3 \\ n_{β_{1},α_{1}} − 1 \end{matrix} ) − T_{i}(β_{1})T_{j}(β_{1}) ( \begin{matrix}n_{β,α} − 3 \\ n_{β_{1},α_{1}} \end{matrix} )] .

If n_{β,α} ≥ 0, then for any k ∈ {1, ⋯, r} the relation R(k) holds:
(R(k))  T_{i}(β)T_{j}(β)a_{k}N_{β,α} = (T_{i}(β)T_{j}(β) − g_{ij}(a_{k} − 1)^{2}) N_{β,α−[k]}

+ ∑_{⊢(β,α−[k])≠0} N_{β_{1},α_{1}}N_{β_{2},α_{2}} ( ∑_{a,b=1}^{p} T_{a}(β_{1})g^{ab}T_{b}(β_{2}) − ∑_{s=1}^{r} b_{s}c_{s})

(T_{j}(β_{1})T_{i}(β_{2})b_{k}c_{k} − T_{i}(β_{1})T_{j}(β_{1})c_{k}^{2}) ( \begin{matrix}n_{β,α−[k]} − 1 \\ n_{β_{1},α_{1}} \end{matrix} ) .

Furthermore, the properties (P1), (P2) and (P3) hold.

(P1) N_{0,α} = 0 unless α = −[i] for some i.

(P2) N_{β,α} = 0 if β is effective and any of the a_{i} is negative.

(P3) N_{β,α} = N_{β,α_{σ}} for any permutation σ of {1, ⋯, r}.

Proof: Lemma 3.1 implies the equation (i). Bceause N_{0,−[i]_{r}} counts the number of curves in the exceptional class E_{i}, N_{0,−[i]_{r}} = 1. So it is sufficient to prove the two recursive relations and the properties (P1), (P2) and (P3). One first want to prove the recursive relations.

To exploit (R_{i,j,m,m}), note that N_{β,α} appears in Φ_{mmm} in the term

N_{β,α} \frac{t_{m}^{n_{β,α}−3}}{(n_{β,α} − 3)!}q^{β}q_{1}^{a_{1}} ⋯ q_{r}^{a_{r}}.

This suggests looking at terms containing t_{m}^{n_{β,α}−3} q^{β}q_{1}^{a_{1}} ⋯ q_{r}^{a_{r}} in RHS of (R_{i,j,m,m}). In ∑_{a,b=1}^{p}(Φ_{jma}g^{ab}Φ_{bim} − Φ_{ija}g^{ab}Φ_{bmm}), the terms without t_{1}, ⋯, t_{m−1} are

∑_{a,b=1}^{p} ∑_{⊢(β,α)} [\frac{T_{j}(β_{1})T_{a}(β_{1})g^{ab}T_{i}(β_{2})T_{b}(β_{2})}{(n_{β_{1},α_{1}} − 1)!(n_{β_{2},α_{2}} − 2)!} − \frac{T_{i}(β_{1})T_{j}(β_{1})T_{a}(β_{1})g^{ab}T_{b}(β_{2})}{n_{β_{1},α_{1}}!(n_{β_{2},α_{2}} − 3)!}]

N_{β_{1},α_{1}}N_{β_{2},α_{2}}q^{β}q_{1}^{a_{1}} ⋯ q_{r}^{a_{r}}.

In ∑_{s=1}^{r}(Φ_{ij(s+p)}Φ_{(s+p)mm} − Φ_{jm(s+p)}Φ_{(s+p)im}), the terms without t_{1}, ⋯, t_{m−1} are

∑_{s=1}^{r} ∑_{⊢(β,α)} [\frac{T_{i}(β_{1})T_{j}(β_{1})b_{s}c_{s}}{n_{β_{1},α_{1}}!(n_{β_{2},α_{2}} − 3)!} − \frac{T_{i}(β_{1})T_{j}(β_{2})b_{s}c_{s}}{(n_{β_{1},α_{1}} − 1)!(n_{β_{2},α_{2}} − 2)!}]
N_{β_{1},α_{1}}N_{β_{2},α_{2}}q^{β}q_{1}^{a_{1}} ⋯ q_{r}^{a_{r}}.

Therefore, if we choose i, j ∈ {1, ⋯, p} such that g_{ij} ≠ 0, then we have

(R(m)^{∗})  N_{β,α} =

\frac{1}{g_{ij}} ∑_{⊢(β,α)≠0} N_{β_{1},α_{1}}N_{β_{2},α_{2}} ( ∑_{a,b=1}^{p} T_{a}(β_{1})g^{ab}T_{b}(β_{2}) − ∑_{s=1}^{r} b_{s}c_{s})

[T_{i}(β_{1})T_{j}(β_{2}) ( \begin{matrix}n_{β,α} − 3 \\ n_{β_{1},α_{1}} − 1 \end{matrix} ) − T_{i}(β_{1})T_{j}(β_{1}) ( \begin{matrix}n_{β,α} − 3 \\ n_{β_{1},α_{1}} \end{matrix} )] .

For k ∈ {1, ⋯, r}, from the relation (R_{i,j,k+p,k+p}) above, we get immediately ( for n_{β,α} ≥ 1) the following recursion formula

(R(k)^{∗})  (g_{ij}a_{k}^{2} − T_{i}(β)T_{j}(β))N_{β,α}

= ∑_{⊢(β,α)} N_{β_{1},α_{1}}N_{β_{2},α_{2}} ( ∑_{a,b=1}^{p} T_{a}(β_{1})g^{ab}T_{b}(β_{2}) − ∑_{s=1}^{r} b_{s}c_{s})

(T_{j}(β_{1})T_{i}(β_{2})b_{k}c_{k} − T_{i}(β_{1})T_{j}(β_{1})c_{k}^{2}) ( \begin{matrix}n_{β,α} − 1 \\ n_{β_{1},α_{1}} \end{matrix} )

Now we want first to prove property (P1). If N_{0,α} ≠ 0, then (β, α) is effective and therefore α ≤ 0. If n_{0,α} = 0, we get α = −[i] for some i ∈ {1, ⋯, r}. If n_{0,α} > 0, we apply (R(k)^{∗}) for an k with a_{k} ≠ 0. We will see that all summands on RHS are divisible by T_{j}(β_{1}) = 0, and thus (P1) follows.

Now we want to prove that the sum in (R(m)^{∗}) only runs over ⊢ (β, α) > 0. It will be shown that the terms with β_{1} = 0 or β_{2} = 0 vanish. Since all summands are divisible by T_{i}(β_{1}), only the case β_{2} = 0 needs be considered. By (P1), either N_{0,γ} = 0 or γ = −[i]. In the second case, both binomial coefficients vanish. Thus relation (R(m)) follows.

Now we want to show relation (R(k)) holds. We first apply relation (R(k)^{∗}) to N_{β,α−[k]}. Then we have

(R(k)^{∗∗})  (g_{ij}(a_{k} − 1)^{2} − T_{i}(β)T_{j}(β))N_{β,α−[k]}

= ∑_{⊢(β,α−[k])} N_{β_{1},α_{1}}N_{β_{2},α_{2}} ( ∑_{a,b=1}^{p} T_{a}(β_{1})g^{ab}T_{b}(β_{2}) − ∑_{s=1}^{r} b_{s}c_{s})

(T_{j}(β_{1})T_{i}(β_{2})b_{k}c_{k} − T_{i}(β_{1})T_{j}(β_{1})c_{k}^{2}) ( \begin{matrix}n_{β,α−[k]} − 1 \\ n_{β_{1},α_{1}} \end{matrix} )

All summands on RHS of (R(k)^{∗∗}) are divisible by T_{j}(β_{1}), thus all nonvanishing summands have C_{1}(β_{1}) > 0. By (P1), N_{0,γ} can be zero only if γ = −[l] for some l ∈ {1, ⋯, r}. Since RHS of (R(k)^{∗∗}) is divisible by c_{k}, the only nonzero summand on RHS of (R(k)^{∗∗}) with β_{2} = 0 occurs for (β_{2}, α_{2}) = (0, −[k]) and is −T_{i}(β)T_{j}(β)a_{k}N_{β,α}. Therefore, it is easy to see from (R(k)^{∗∗}) that (R(k)) holds.

The remain proof is similar to that of Theorem 3.6 of [GP]. We omit it now.

4 Functoriality property for a blowup

Quantum cohomology is an admixture of homology and cohomology. Therefore, it is too much for us to expect that quantum cohomology has the same functoriality property as ordinary cohomology. Crauder and Miranda[CM] first consider the case of rational surfces and showed that there is some degree of functoriality for a general blowup. In this section, we will consider blowups of smooth projective manifolds of dimension 2n instead of surfaces and show that the big quantum cohomology has some degree of functoriality for a blowup.

The fundamental reason why cohomology is a very useful is its naturality. Namely, a continuous map induces a ring homomorphism on cohomology. Therefore, the fundamental problem in quantum cohomology is the quantum naturality problem[R2]: Define “morphism” of symplectic manifolds so that quantum cohomology is natural. Qin and Ruan [QR] showed that quantum cohomology is not natural for fibrations. Their results also shows that possible “morphisms” must be very rigid. The existence of these rigid morphisms will set apart quntum cohomology from ordinary cohomology and gives it its own identity.

To define a homomorphism of the quantum product, we need to match Novikov rings and hence symplectic classes. Here we view quantum cohomology as a theory of GW invariants instead of just quantum product. Since the GW potential Φ contains the same information as quantum product, therefore, it is more convenient to work directly with the GW potential Φ.

Definition4.1: Suppose X, Y are symplectic manifolds and that f :

X ⟶ Y is a map and induces group homomorphism

f_{∗} : H_{2}(X, Z) ⟶ H_{2}(Y, Z) and f^{∗} : H^{∗}(Y, R) ⟶ H^{∗}(X, R).

We say f is weakly natural with repect to the big quantum cohomology if Φ^{X}(f^{∗}γ) = Φ^{Y} (γ) after change of formal variable β ⟶ f_{∗}(β). We say f is natural with repect to the big quantum cohomology if Φ^{X}(f^{∗}γ) = Φ^{Y} (γ) after change of formal variable q^{β} ⟶ q^{f_{∗}(β)}. If f_{∗} and f^{∗} are also isomorphisms and f is natural, we say f induces an isomorphism on big quantum cohomology or X and Y have the same big quantum cohomology.

As in [R2], if necessary, we will consider two power series F, G as the same if F = H + F′, G = H + G′ are power series expansion for the same rational function at the different points. For example, we can expand \frac{1}{1−t} = ∑_{i=0} t^{i} at t = 0 or \frac{1}{1−t} = \frac{1}{−t(1−t^{−1})} = − ∑_{i=0} t^{−i−1} at t = ∞. Hence, we need to treat ∑_{i=0} t^{i}, ∑_{i=0} t^{−i−1} as the same power series.

Theorem 4.2: Suppose that X is a smooth projective manifold of dimension 2n and p : X̃_{r} ⟶ X is a blowup at r general points. Then p is weakly natural with respect to big quantum cohomology.

Proof: Without loss of generalization, we assume r = 1. For simplicity, we also assume that H^{odd}(X) = 0. Let T_{0} = 1, ⋯, T_{p}, ξ, ⋯, ξ^{n−1}, T_{m} be a basis of H^{∗}(X̃), where T_{0} = 1, T_{1}, ⋯, T_{m} is a basis of H^{∗}(X) and ξ is a generator of the cohomology H^{∗}(E) of the exceptional divisor E. Let γX̃ = ∑_{i=0}^{m} t_{i}T_{i} + ∑_{i=1}^{n−1} y_{i}ξ^{i} and γ^{X} = ∑_{i=0}^{m} t_{i}T_{i}. Denote by (β, α) the homology class β − αe ∈ H_{2}(X̃), where e is the class of a line in the exceptional divisor E. Then we have GW potentials of X̃ and X as follows:

ΦX̃(t_{0}, t_{1}, ⋯, t_{m}, y_{1}, ⋯, y_{n−1})

= \frac{1}{6} ∫X̃ (γX̃)^{3} + ∑_{n≥3} ∑_{(β,α)∈H_{2}(X̃,Z)−\{0\}} \frac{1}{n!}ΨX̃ ; (β,α)((γX̃)^{n})q^{β}q_{1}^{α}.

Φ^{X}(t_{0}, t_{1}, ⋯, t_{m}) = \frac{1}{6} ∫_{X} (γ^{X})^{3} + ∑_{n≥3} ∑_{β∈H_{2}(X,Z)−\{0\}} \frac{1}{n!}Ψ_{β}^{X} ((γ^{X})^{n}) q^{β}.

Therefore, we have

ΦX̃(p^{∗}γ^{X}) = \frac{1}{6} ∫X̃ (p^{∗}γ^{X})^{3} + ∑_{n≥3} ∑_{(β,α)∈H_{2}(X̃,Z)−\{0\}} \frac{1}{n!}ΨX̃ ; (β,α) ((p^{∗}γ^{X})^{n}) q^{β}q_{1}^{α}.

By the degree reason and Lemma 3.1, after replacing (β, α) by p_{∗}(β, α) = β, we actually have the terms with q_{1}^{α}, (α ≠ 0) must be zero. Therefore, we have ΦX̃(p^{∗}γ^{X}) = Φ^{X}(γ^{X}). This completes the proof of the theorem.

Remark 4.3: Replacing γX̃ by p^{∗}γ^{X} is equivalent to seting y_{1} = ⋯ = y_{n−1} = 0 in ΦX̃(t_{i}, y_{j}). The change of formal variable (β, α) ⟶ p_{∗}(β, α) = β is also equivalent to truncating the terms with q_{1}^{α}(α ≠ 0) in ΦX̃(p^{∗}γ^{X}). The real meaning of our theorem means that there is some degree of functoriality of the big quantum cohomology.

Example: Suppose that X is a smooth projective surfaces as in section 3. If we set ϵ = t_{0}T_{0} + t_{m}T_{m}, by the equality (3.1), then the big quantum product of p^{∗}T_{i} and p^{∗}T_{j} is given by

p^{∗}T_{i} ∗ p^{∗}T_{j} = ∑_{l,k=1}^{p} ∑_{n=0}^{∞} ∑_{(β,α)} \frac{1}{n!}ΨX̃ ; (β,α)(T_{i}, T_{j}, T_{l}, ϵ^{n})e^{∑_{i=1}^{p} t_{i} ∫_{β} T_{i}}e^{t_{p+1}α}q^{β}q_{1}^{α}g^{lk}T_{k}

− ∑_{k=p+1}^{p+r} ∑_{n=0}^{∞} ∑_{(β,α)} \frac{1}{n!}ΨX̃ ; (β,α)(T_{i}, T_{j}, T_{k}, ϵ^{n})e^{∑_{i=1}^{p} t_{i} ∫_{β} T_{i}}e^{t_{p+1}α}q^{β}q_{1}^{α}T_{k}

+ (T_{i} · T_{j})T_{m} + ∑_{n=0}^{∞} \frac{1}{n!}ΨX̃ ; (β,α)(T_{i}, T_{j}, T_{m}, ϵ^{n})e^{∑_{i=1}^{p} t_{i} ∫_{β} T_{i}}e^{t_{p+1}α}q^{β}q_{1}^{α}T_{0}.

On the other hand, we have

T_{i} ∗ T_{j} = ∑_{l,k=1}^{p} ∑_{n=0}^{∞} ∑_{β} \frac{1}{n!}Ψ_{β}^{X}(T_{i}, T_{j}, T_{l}, ϵ^{n})e^{∑_{i=1}^{r} t_{i} ∫_{β} T_{i}}q^{β}g^{lk}T_{k}
+ (T_{i} · T_{j})T_{m} + ∑_{n=0}^{∞} ∑_{β} \frac{1}{n!}Ψ_{β}^{X}(T_{i}, T_{j}, T_{m}, ϵ^{n})e^{∑_{i=1}^{r} t_{i} ∫_{β} T_{i}}q^{β}T_{0}.

Therefore, if we identify T_{i} with p^{∗}T_{i}, it is easy to see that

p^{∗}T_{i} ∗X̃ p^{∗}T_{j}|_{q_{1}=y_{1}=0} = T_{i} ∗_{X} T_{j}.

Therefore, we may consider the quantum product of T_{i} and T_{j} on X is the restriction of the quantum product on X̃ of their pullbacks to H^{∗}(X, R). Our theorem also shows that there is some degree (i.e. module the effect of homology) of functoriality property of the big quantum cohomology for a blowup. In the case of rational surfaces, our conlusion coincides with that of [CM].

References

[CM] B. Crauder, R. Miranda, Quantum cohomology of rational surfaces, in The moduli space of curves (R. Dijkgraaf, C. Faber and G. van der Geer, eds) Birkhauser, Basel,1995, 33-80.

[DVV1] R. Dijkgraaf, E. Verlinde, H. Verlinde, Topological strings in d < 1, Neuclear Phys. B 352(1991), 59-86.

[DVV2] R. Dijkgraaf, E. Verlinde, H. Verlinde, Notes on topological string theory and 2D gravity, in String theory and quantum gravity, Proceedings of the Trieste Spring school, 1990, (M. Geen et al, eds), Wrold Scientific, Singapore, 1991,91-156.

[GP] L. Göttsche, R. Pandharipande, The quantum cohomology of blowups of P^{2} and enumerative geometry, J. Diff. Geom., 48(1998), 61-90.

[H1] J. Hu, Gromov-Witten invariants of blowups along points and curves, Math. Z. 233(2000), 709-739. math.AG/9810081,

[KM] M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164(1994), 525-562.

[LVW] W. Lerche, C. Vafa, N. Warner, Nuclear Phys. B,324(1989),427.

[QR] Z. Qin, Y. Ruan, Quantum cohomology of projective bundles over P^{2}, Trans. AMS, 350(1998), 3615-3638.

[RT] Y, Ruan, G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom., 42(1995), 259-367.

[R1] Y. Ruan, Quantum cohomology and its applications, Lecture at ICM98, Doc. Math.(1998) Extra vol. II, 411-420.

[R2] Y. Ruan, Surgery, quantum cohomology and birational geometry, in Northern California Symplectic Geometry Seminar(Y. Eliashberg, D. Fuchs, T. Ratiu, A. Weinstein, eds), AMS Translations, Series 2, vol. 196(1999), 183-198.

[W1] E. Witten, Topological sigma models, Comm. Math. Phys.118(1988), 411-449.

[W2] E. Witten, On the structure of the topological phase of twodimensional gravity, Nuclear Phys. B 340(1990), 281-332.

Email address: stsjxhu@zsu.edu.cn