ELLIPTIC GW INVARIANTS OF BLOWUPS ALONG CURVES AND SURFACES

We established a relation between elliptic Gromov-Witten invariants of a symplectic manifold M and its blowups along smooth curves and surfaces.


Introduction
Over the last few years, many mathematicians contributed their efforts to establish the mathematical foundation of the theory of quantum cohomology or Gromov-Witten (GW) invariants.In 1995, Ruan and Tian [13,15] first established for the semipositive symplectic manifolds.Recently, the semipositivity condition has been removed by many authors.Now, the focus turned to the calculations and applications.Many Fano manifolds were computed.We think that it is important to study the change of GW invariants under surgery.
Some recent research indicated that there is a deep amazing relation between quantum cohomology and birational geometry.The quantum minimal model conjecture [14] leads to attempt to find quantum cohomology of a minimal model without knowing minimal model.This problem requires a thorough understanding of blowup type formula of GW invariants and quantum cohomology.
According to McDuff [11], the blowup operation in symplectic geometry amounts to a removal of an open symplectic ball followed by a collapse of some boundary directions.Lerman [8] gave a generalization of blowup construction, "the symplectic cut." Let M be a compact symplectic manifold of dimension 2n, M the blowup of M along a smooth submanifold.Denote by p : M → M the natural projection.Denote by Ψ M (A,g) (α 1 , ...,α m ) the genus g GW invariant of M. The authors refer the interested reader to [15] for the definition of GW invariants.
In [5,6], we mainly concentrated on the changes of genus zero GW invariants under blowup along smooth curves and surfaces.In this paper, we mainly concentrate on the elliptic GW invariants.These invariants were first discussed in [1] and since then have been studied in various contexts, see [3,12].An elegant recursion was predicted in [2] using the method of Virasoro constraints.In this paper, we will generalize some results about the cases of blowing up along curves and surfaces to the case of genus-one.Our main results are the following.
Theorem 1.1.Suppose that C is a smooth curve in M such that either its genus g 0 ≥ 2 or g 0 ≤ 1 and satisfy either degα i = 1 or degα i = 1 and support away from C.Then, About the changes of GW invariants of blow up of symplectic manifold along a smooth surface, in this paper, we will assume that the symplectic manifold is semipositive.Our result is the following theorem.
Theorem 1.2.Suppose that M is a semipositive compact symplectic manifold and S is a smooth surface in M.

General review of gluing formula
The proof of our result is an application of the gluing formula for symplectic cutting developed by Li and Ruan [9].In algebraic geometry, Li [10] proved a completely analogous degeneration formula of GW invariants.Another symplectic version of the gluing formula of GW-invariants is due to Ionel and Parker [7].
Suppose that H : M 0 → R is a proper Hamiltonian function such that the Hamiltonian vector field X H generates a circle action, where M 0 ⊂ M is an open domain.By adding a constant, we may assume that 0 is a regular value.Then, H −1 (0) is a smooth submanifold preserved by the circle action.The quotient Z = H −1 (0)/S 1 is the famous symplectic reduction.Namely, it has an induced symplectic structure.We can cut M along H −1 (0).Suppose that we obtain two disjoint components M ± which have the boundary H −1 (0).We can collapse the S 1 -action on H −1 (0) to obtain M ± containing a real codimension two submanifold Z = H −1 (0)/S 1 , see [9, Section 2 and pages 156-157] for details.There is a map where M+ ∪ Z M− is the union of M± along Z.
To formulate the gluing formula, we need the terminology of relative GW invariants.Here, we copy the definition of [9, Section 4 and page 157].We define a relative GW invariant Ψ (M,Z) by counting the number of relative stable holomorphic maps intersecting Z at finitely many points with prescribed tangency.Let T m = (t 1 ,...,t m ) be a set of nonnegative integers such that i t i = Z * (A), where Z * is the Poincare dual of Z and A ∈ H 2 (M).We order them such that t 1 = ••• = t l = 0 and t i > 0 for i > l.Consider the moduli space ᏹ A (g,T m ) of genus g pseudoholomorphic maps f such that f has marked points (x 1 ,...,x m ) with the property that f is tangent to Z at x i with order t i .Here, t i = 0 means that there is no intersection.Then, we compactify ᏹ A (g,T m ) by ᏹ(g,T m ), the space of relative stable maps.We have evaluation map for i ≤ l and for j > l.Roughly, the relative GW-invariants are defined as Π i e * i α i ∧ Π j e * j β j . (2.4) ) be J-holomorphic curves such that u + and u − have ν ends and they converge to the same periodic orbits at each end.Suppose that Σ ± have m ± marked points, respectively.Here, Σ ± may not be connected, see [9,Section 4].Suppose that Σ = Σ + Σ − has genus g and [u(Σ)] = A. If we consider the index of the operators D u ± = D∂ J u ± , see [9,Section 4] for its definition, then we have the following proposition.
The following gluing formula holds: where β i δ i, j β j is associated to every intersection with Z and denote the product of relative invariants corresponding to each component.

Proof of main theorem
Proof of Theorem 1.1.Since C is a smooth curve in M, the normal bundle N C is a symplectic vector bundle.By symplectic neighborhood theorem, there is a tubular neighborhood ᏺ δ (C) of C which is symplectomorphic to the normal bundle N C .We perform the symplectic cutting as in [5, Section 2.1].We obtained From the divisor property, the skew symmetry of GW invariants and our assumptions, if we choose a sufficiently small δ > 0, without loss of generality, we may assume α + i = 0. Similar to the proof of [5, Theorem 1.2], by the gluing formula of GW invariant, we first consider the contribution of each component to the GW invariants.Therefore, we consider the gluing component where K ± = (0,...,0,k 1 ,...,k ν ) with m ± many zeros, respectively.From Proposition 2.1, we have since g = 1 in our case.We will use the same convention as [5].We assume that u ± : Σ ± → M ± may have l ± connected components u ± i : Σ ± i → M ± , i = 1,...,l ± .Suppose Σ ± i have arithmetic genus g ± i , g ± = Σg ± i , with m ± i marked points and m ± = Σm ± i .From [5, Remark 2.3], it is not difficult to see that ū+ i can be identified as a stable J-holomorphic curve h + i in M + .Then, from [5, Proposition 2.4], we have Now, we want to calculate C 1 [h + i ] in two cases of our theorem.Case 1.The genus g 0 ≥ 2.
In this case, we claim that all stable J-holomorphic maps h + i can only stay in fibers of M + = P(N C ⊕ ᏻ).Otherwise, suppose that there is a stable J-holomrphic curve h + i : Σ → M + which does not stay in a fiber.Denote by π : P(N C ⊕ ᏻ) → C the projection of the projective bundle.Then, we have a stable J-holomorphic map π We can perform pregluing as in [9, Section 5, pages 208-209] and obtain a system of small perturbed J-holomorphic curves f n : Σ n → C which represent the class [π • h + i ] and satisfy the perturbed Cauchy-Riemann equation Σ n is a smooth Riemann surface.Actually, we can choose ν n → 0 as n → ∞.Therefore, by Gromov compactness theorem, we have that f n weakly converges to a (possibly reducible) J-holomorphic curve u = (u 1 ,...,u N ) and [π • h + i ] = N j=1 [u j ] = 0. Therefore, we have a nonconstant J-holomorphic curve f : Σ 1 → C and Σ 1 has genus less than 2. It is well known that if f : S → S is a holomorphic map between compact Riemann surfaces, then the genus of S and S satisfies g(S) ≥ g(S ) unless f is constant (see [4, page 219]).Since g(C) = g 0 ≥ 2, we have a contradiction.So, our claim is true.
A simple index calculation [5] shows that C 1 [h + i ] = n k j , where summation runs over ends of component u + i .In this case, we have nξ, here ξ is the class of infinite section in P(N C ⊕ ᏻ) over C. Therefore, from our positive assumption, an intersection multiplicity calculation shows that In this case, we have Summarizing the above two cases, from (3.4), we have This implies Ψ C = 0 except m − = m.So, we may assume m − = m.We also may assume Otherwise, Theorem 1.1 follows from the degree reason.Therefore, we have Therefore, by the definition of relative GW invariants, we have for any β b ∈ H * (Z), So, now it remains to show that To prove this, we perform the symplectic cutting for M around the exceptional divisor E. Therefore, we have Now, we use the gluing formula to prove the contribution of stable J-holomorphic curves in M which touch the exceptional divisor E to the GW invariants of M is zero.We consider the gluing component Since We assume that u ± : Σ ± → M ± has l ± connected components u ± i : Σ ± i → M ± , i = 1,..., l ± .From [5, Remark 2.3], it is not difficult to see that ū+ i can be identified as stable Jholomorphic curve h + i in M.Then, from [5, Proposition 2.4], we have where C 1 is the first Chern class of M+ .Let V be a complex rank r vector bundle over X, and π : P(V ) → X the corresponding projective bundle.Let ξ V be the first Chern class of the tautological line bundle in P(V ).A simple calculation shows that (3.18) Note that M+ = P(N E ⊕ ᏻ) and E = P(N C ). Applying (3.18) to M+ and E, we obtain J. Hu and H.-Y. Zhang 87 where ξ 1 and ξ are the first Chern classes of the tautological line bundles in P(N C ) and P(N E ⊕ ᏻ), respectively.Here, we denote Chern class and its pullback by the same symbol.
It is well known that the normal bundle to E in M is just the tautological bundle on E ∼ = P(N C ).Therefore, C 1 (N E ) = ξ 1 .So, we have We know that M is a projective bundle over E with fiber P 1 .Let L be the class of a line in the fiber P 1 and e the class of a line in the fiber P n−2 in E = P(N C ). Denote by [h + i ] C the homology class of the projection in C of the curve , where the summation runs over ends of u + i , and For Case 1, we have [h + i ] C = 0. Therefore, we have Plugging in (3.17), we have From the degree reason, we also may assume since ν > 0, k i > 0. Therefore, by the definition of relative GW invariants, we have for any Therefore, the contribution of J-holomorphic curves to the GW invariant is nonzero only if it does not touch the exceptional divisor E, that is, C = {p!(A) − ,1,m}.Therefore, we have Proof of Theorem 1.2.Since S is a smooth surface, the normal bundle N S is a symplectic vector bundle.By symplectic neighborhood theorem, there is a tubular neighborhood ᏺ δ (S) of S which is symplectomorphic to the normal bundle N S .We perform the symplectic cutting as in [5, Section 2.1].We obtain We may assume α + i = 0 if we choose a sufficiently small δ > 0 because of the assumption of α i .
Similar to the proof of Theorem 1.1, we first consider the contribution of each gluing component to the GW-invariants.Therefore, we consider the component (3.2).From Proposition 2.1, we have (3.3).
We assume u ± : Suppose Σ ± i has arithmetic genus g ± i , g ± = g ± i with m ± marked points and m ± = m ± i .The similar calculation to that in the proof of Theorem 1.1 shows that (3.30) Therefore, we have The same argument as in the proof of Theorem 1.1 shows that the contribution of the component C to the GW-invariant of M is nonzero only if C has (3.16).We also assume degα i = 2C 1 (A) + 2m. (3.32) The same argument as in the proof of Theorem 1.1 shows that there is a symplectic cutting such that Ψ C = 0 except C = {A − ,1,m}.From the gluing theorem, we have To prove this, we perform the symplectic cutting for M around E as in the proof of Theorem 1.1.Therefore, we have (3.14).
We also use the gluing theorem to prove that the contribution of stable J-holomorphic curves in M which touch the exceptional divisor E to the GW invariant of M is zero.We consider the component (3.15).
Since α + i = 0, 1 ≤ i ≤ m, we have Ψ C = 0 except (3.16).Similar calculation to that in the proof of Theorem 1.1 shows that Ind D u + = 2(n − 3) l + − g + + 2ν + 2(2n − 5) k i , The same argument as in the proof of Theorem 1.1 shows (3.28).The rest of the proof is the same as that of the proof of Theorem 1.1.So, we omit it.This completes the proof of Theorem 1.2.

) where C 1
is the first Chern class of M and m = m + + m − .Suppose that the homology classes of u + , u − , u are A + , A − , A, respectively.If (u + ,u − ) is another representative and glue to u , by [9, Lemma 2.11], we have [u ] = [u].Denote by C = {A + ,g + ,K + ;A − ,g − ,K − } the gluing component, where K ± = (0,...,0,k 1 ,...,k ν ) with m ± zeros.The gluing formula of Li and Ruan counted the contribution of the gluing components to GW invariant of M. Denote by Ψ C the contribution of C.
.16) From Proposition 2.1, we have (3.3),where C 1 denotes the first Chern class of M.
M − and p!(A) is a nonexceptional class, we may identify the homology class p!(A) − with A − .Hence, Theorem 1.1 follows.