High genus GW invariants of blowups along curves and surfaces

Jianxun Hu

Department of Mathematics, Zhongshan University, Guangzhou, 510275, P. R. China

Hou-yang Zhang

The Science College, Northern Jiaotong University, Beijing, 100044, P. R. China

Abstract:

In this paper, we established a relation between high genus Gromov-Witten invariants of a symplectic manifold M and its blowups along smooth curves and surfaces.

Keywords: Gluing formula, Blowup, Log GW-invariant.

2000 AMS Classification: Primary 53D45. Secondary 14N35.

1 Introduction

Over the last a few years, many mathematicians contributed their efforts to establish the mathematical foundation of the theory of quantum cohomology or GW(Gromov-Witten) invariants. In 1995, Ruan-Tian [R1, RT] first established for the semipositive symplectic manifolds. Recently the semipositivity condition has been removed by many authors. Now the focus is on 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 [R2, R3] 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 [M] the blowup operation in symplectic geometry amounts to a removal of an open symplectic ball followed by a collapse of some boundary directions. Lerman [L] gave a generalization of blowup construction, “the symplectic cut”.

Let M be a compact symplectic manifold of dimension 2n, M̃ be the blowup of M along a smooth submanifold. Denote by p : M̃ ⟶ M the natural projection. Denote by Ψ_{(A,g)}^{M}(α_{1}, ⋯, α_{m}) the genus g GW invariant of M. The authors refer the interested reader to [RT] for the definition of GW invariants.

In [H1, H2], we considered the changes of genus zero GW invarints under blowup along a smooth curves and surfaces. In this paper, We will generalize some results about the cases of blowing up along curves and surfaces to the case of any genus. If the blown-up submanifolds are smooth curves with nonzero genus or special genus zero curves, we can show the following theorems:

Theorem 1: Suppose that C is a smooth curve in M such that either its genus g_{0} ≥ 1 or g_{0} = 0 and C_{1}(M)(C) ≥ 0, where C_{1}(M) denotes the first Chern classes of M and its normal bundle respectively. A ∈ H_{2}(M), α_{i} ∈ H^{∗}(M), 1 ≤ i ≤ m. Then

Ψ_{(A,g)}^{M}(α_{1}, …, α_{m}) = ΨM̃ ; (p!(A),g)(p^{∗}α_{1}, …, p^{∗}α_{m})

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

Theorem 2: Suppose that M is a semipositive compact symplectic manifold and S is a smooth surface in M. If A ∈ H_{2}(M), α_{i} ∈ H^{∗}(M), 1 ≤ i ≤ m, satisfy either degα_{i} > 2 or degα_{i} ≤ 2 and support away from S. Then

Ψ_{(A,g)}^{M}(α_{1}, ⋯, α_{m}) = ΨM̃ ; (p!(A),g)(p^{∗}α_{1}, ⋯, p^{∗}α_{m}),

where p!(A) = P Dp^{∗}P D(A).

Acknowledgement: The first author would like to thank Prof. Yongbin Ruan for his many suggestive discussion and encouragement. The first author also thanks Max-Planck-Institut für Mathematik in Bonn for providing an excellent research environment and support.

2 Review of Gluing Formula

The proofs of our results depend strongly on the gluing formula for symplectic cutting developed by Li and Ruan [LR]. Let \overline{M}^{+}, \overline{M}^{−} be the symplectic cuttings of M along N ⊂ M. Here N is the common boundary of M^{±} with a S^{1}-action and a generic level set of a local periodic hamiltonian function. The quotient Z = N/S^{1} is the famous symplectic reduction, see [LR], p9-10 for details. Li and Ruan defined Log GW invariants Ψ^{(\overline{M}^{±},Z)} of \overline{M}^{±} relative Z by counting the number of log stable holomorphic maps intersecting Z at finitely many points with prescribed tangency. In this section, we want to extend Li and Ruan’s gluing formula to the case with more ends. Li and Ruan have pointed out that their proof works in this case, see Remark 5.8 of [LR]. Therefore, here we only state the results.

Let Z_{i}, i = 1, ⋯, l, be real codimension two symplectic submanifolds of M. By a result of Guillemian and Sternberg [GS], the symplectic structure of each Z_{i} is modeled on a neighborhood of Z in either \overline{M}^{+} or \overline{M}^{−}.

Let K = {0, ⋯, 0|k_{1}^{(1)}, ⋯, k_{ν_{1}}^{(1)}| ⋯ |k_{1}^{(l)}, ⋯, k_{ν_{l}}^{(l)}} with m-many zeros be a set of nonnegative integers with k_{j}^{i} > 0. Denote by M_{A}(g, Z_{1}, ⋯, Z_{l}, K) the space of genus g pseudo-holomorphic maps u, which represent the homology class A, such that u has marked points (x_{1}, ⋯, x_{m}, x_{m+1}, ⋯, x_{m+ν_{1}}, ⋯, x_{m+∑_{k=1}^{l} ν_{k}}) with the property u is tangent to Z_{i} at x_{m+∑_{k=1}^{i−1} ν_{k}+j} with order k_{j}^{i}

and f(x_{j}) ∉ Z_{i} for 0 ≤ j ≤ m and ∑_{j=1}^{ν_{i}} k_{j}^{(i)} = Z_{i}^{∗}(A), where Z_{i}^{∗} is the Poincare dual to Z_{i}. We compactify M_{A}(g, Z_{1}, ⋯, Z_{l}, K) by \overline{M}_{A}(g, Z_{1}, ⋯, Z_{l}, K), the space of log stable maps. We have evaluation maps

e_{k} : \overline{M}_{A}(g, Z_{1}, ⋯, Z_{l}, K) ⟶ M,

for marked points x_{1}, ⋯, x_{m} and

e_{j}^{i} : \overline{M}_{A}(g, z_{1}, ⋯, Z_{l}, K) ⟶ Z_{i}, i = 1, ⋯, l, j = 1, ⋯, ν_{i},

for marked points x_{m+∑_{k=1}^{i−1} ν_{k}+j}. Roughly the Log GW invariants may be defined as

Ψ_{(A,g,K)}^{(M,Z_{1},···,Z_{l})}(α_{1}, ⋯, α_{m}, β_{j}^{i}) = ∫_{\overline{M}_{A}(g,Z_{1},···,Z_{l},K)} Π_{k=1}^{m}e_{k}^{∗}α_{k} ∧ Π_{i,j}(e_{j}^{i})^{∗}β_{j}^{i},

for α_{i} ∈ H^{∗}(M), β_{j}^{i} ∈ H^{∗}(Z_{i}). To be precise, we need to construct a virtual neighborhood of \overline{M}_{A}(g, Z_{1}, ⋯, Z_{l}, K). Then we take the integrand over the normalization of the virtual neighborhood.

Suppose Z is a common real codimension two symplectic submanifold of \overline{M}^{±}. Next we will state the gluing formula of Log GW invariants with many ends. Clearly other ends do not influence our gluing when we are gluing along some end Z. Therefore, to simplify the notations, we just drop data from other ends when we write the gluing formula for some end Z

Given u^{±} ∈ M_{A^{±}}(\overline{M}^{±}, Z, Z_{1}^{±}, ⋯, Z_{l^{±}}^{±}, K^{±}) where K^{±} have m^{±}-many zeros respectively and m = m^{+} + m^{−}. We consider the linearization of \overline{∂}-operator

D_{u^{±}} = D∂_{J}(u^{±}) : C^{∞}((u^{±})^{∗}T M^{±}) ⟶ Ω^{0,1}((u^{±})^{∗}T M^{±}).

If we choose a proper weighted Sobolev norm, we can define the Fredholm index IndD_{u^{±}}. see section 3.4 of [LR] for details. We also can glue u^{+} and u^{−} along the end Z to obtain a pseudo-holomorphic map u in M. Suppose that [u] = A. Then

Proposition 2.1:([LR], Theorem 3.23)

IndD_{u^{+}} + IndD_{u^{−}} = 2(n − 1)ν + 2C_{1}(A) + 2(n − 3)(1 − g) + 2m,

where C_{1} is the first Chern class of 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 Lemma 2.11 of [LR], we have [u′] = [u]. Denote by C = {A^{+}, g^{+}, K^{+}; A^{−}, g^{−}, K^{−}} the gluing component. The gluing formula counted the contribution of the gluing components to Log GW invariant of M. Denote by Ψ_{C} the contribution of C.

Choose a homology bass {β_{b}} of H^{∗}(Z, R). Let (δ_{ab}) be its intersection matrix. There is a map

π : M ⟶ \overline{M}^{+} ∪_{Z} \overline{M}^{−},

where \overline{M}^{+} ∪_{Z} \overline{M}^{−} is the union of \overline{M}^{±} along Z. For the gluing component C = {A^{+}, g^{+}, m^{+}, K^{+}; A^{−}, g^{−}, m^{−}, K^{−}} where K^{±} = (0, ⋯, 0|k_{1}, ⋯, k_{ν}|(k_{j}^{i})^{±}) with m^{±}-many zeros, we have

Proposition 2.2: Let α_{i}^{±} be differential forms with deg α_{i}^{+} = deg α_{i}^{−} even. Suppose that α_{i}^{+}|_{Z} = α_{i}^{−}|_{Z} and α_{i}^{+} ∪_{Z} α_{i}^{−} ∈ H^{∗}(\overline{M}^{+} ∪_{Z} \overline{M}^{−}, R). Let α_{i} = π(α_{i}^{+} ∪_{Z} α_{i}^{−}). We have the following gluing formula

Ψ_{C}^{(M,Z_{i}^{+},Z_{k}^{−})}(α_{1}, ⋯, α_{m}|(β_{j}^{i})^{+}|(β_{j}^{k})^{−}) = |K| ∑_{I,J} Ψ_{(A^{+},g^{+},K^{+})}^{(\overline{M}^{+},Z,Z_{i}^{+})}(α_{i}^{+}, β_{I}, (β_{j}^{i})^{+})
× δ^{I,J}Ψ_{(A^{−},g^{−},K^{−})}^{(\overline{M}^{−},Z,Z_{k}^{−})}(α_{i}^{−}, β_{J}, (β_{j}^{k})^{−}),

where we associate β_{i}δ^{ij}β_{j} to every intersection with Z and put |K| = k_{1} ⋯ k_{ν},

δ^{I,J} = δ^{i_{1},j_{1}} ⋯ δ^{i_{ν},j_{ν}}. Denote by Ψ_{(A^{±},g^{±},K^{±})}^{(\overline{M}^{±},Z,Z_{i}^{±})}(α_{i}^{±}, β_{I}, (β_{j}^{i})^{±}) the product of Log GW invariants corresponding to each component.

This Proposition is an easy extension of Theorem 5.8 of [LR] to Log GW invariant. Their proof works in this case. Therefore we omit the proof.

Proposition 2.3: For C = {A^{±}, g^{±}, (0, ⋯, 0)}, we have

Ψ_{C}(α_{i}^{±}) = Ψ_{(A^{±},g^{±},(0,···,0)}^{(\overline{M}^{±},Z)}(α_{i}^{±}).

3 Proof of main theorem

Proof of Theorem 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 N_{δ}(C) of C which is symplectomorphic to the normal bundle N_{C}. We perform the symplectic cutting as in section 2.1 of [H1]. We obtained

\overline{M}^{+} = P(N_{C} ⊕ O), \overline{M}^{−} = M̃.

¿From the divisor property and skew symmetry of GW-invariants, without loss of generality, we may assume that degα_{i} > 2, 1 ≤ i ≤ m. Therefore, if we choose a sufficiently small δ > 0, we may also assume α_{i}^{+} = 0.

Similar to the proof of Theorem 1.2 of [H1], we first consider the contribution of each component to the GW-invariants. Therefore, we consider the component

C = {A^{+}, g^{+}, m^{+}, K^{+}; A^{−}, g^{−}, m^{−}, K^{−}}.

¿From Proposition 2.1, we have

IndD_{u^{+}} + IndD_{u^{−}} = 2(n − 1)ν + 2C_{1}(A) + 2(n − 3)(1 − g) + 2m.

We will use the same convention as [H1]. We assume u^{±} : Σ^{±} ⟶ M^{±} may have l^{±} connected components u_{i}^{±} : Σ_{i}^{±} ⟶ M^{±}, i = 1, …, l^{±}. Suppose Σ_{i}^{±} have arithemetic genus g_{i}^{±}, g^{±} = Σg_{i}^{±}, with m_{i}^{±} marked points and m^{±} = Σm_{i}^{±}. The same calculation as in the proof of Theorem 1.5 of [H1] shows

IndD_{u^{+}} ≥ 2(n − 3)(l^{+} − g^{+}) + 2(n − 1) ∑ k_{i} + 2ν + 2m^{+},
IndD_{u^{−}} ≤ 2C_{1}(A) + (2n − 6)(1 − l^{+} + g^{+} − g)
−2(n − 1)(ν − ∑ k_{i}) − 2ν + 2m^{−}.

Since α_{i}^{+} = 0, 1 ≤ i ≤ m, if m^{+} > 0, we have for any β_{b} ∈ H^{∗}(Z)

Ψ_{(A^{+},g^{+},m^{+},K^{+})}^{(\overline{M}^{−},Z)}(α_{i}^{+}, β_{b}) = 0.

This implies Ψ_{C} = 0 except m^{−} = m. So we may assume m^{−} = m. We also may assume

∑ degα_{i} = 2C_{1}(A) + 2(n − 3)(1 − g) + 2m.

Otherwise, Theorem 1 follows from the degree reason.

About the contribution of a gluing component, we have the following claim:

Claim: There exists a symplectic cutting such that the contribution Ψ_{C} = 0 excpt C = {A^{−}, g, m}. Here no K^{±} means K^{±} = {0, ⋯, 0}.

First of all, we prove that Ψ_{C} = 0 for the gluing component C with g^{+} = 0 and ν > 0. In fact, we have

∑ deg(α_{i}^{−}) = 2C_{1}(A) + 2(n − 3)(1 − g) + 2m
> 2C_{1}(A) + 2(n − 3)(1 − l^{+} + g^{+} − g)
+2(n − 1)(ν − ∑ k_{i}) − 2ν + 2m
≥ IndD_{u^{−}},

since ν > 0, k_{i} > 0, n ≥ 3. Therefore, by the definition of Log GW-invariants, we have for any β_{b} ∈ H^{∗}(Z)

Ψ_{(A^{−},g^{−},m,K^{−})}^{(\overline{M}^{−},Z)}(α_{i}^{−}, β_{b}) = 0.

Therefore, from Proposition 2.2, we have Ψ_{C} = 0.

Next, we consider the contribution of gluing components with g^{+} > 0. From the connectness of stable maps and the assumption of our theorem, it follows that g^{+} > 0 implies ν > 0.Therefore, we only consider the component C with g^{+} > 0 and ν > 0. By the gluing formula, we have

Ψ_{C} = ∑_{I,J} |K|Ψ_{(A^{−},g^{−},K^{−})}^{(\overline{M}^{−},Z)}(α_{1}^{−}, ⋯, α_{m}^{−}, β_{I})δ^{I,J}Ψ_{(A^{+},g^{+},K^{+})}^{(\overline{M}^{+},Z)}(β_{J}).

We perform the symplectic cutting for \overline{M}^{+} along the curve C. Then the same arguement as above shows

Ψ_{C} = ∑_{I,J} |K|Ψ_{(A^{−},g^{−},K^{−})}^{(\overline{M}^{−},Z)}(α_{1}^{−}, ⋯, α_{m}^{−}, β_{I})δ^{I,J}

× ∑_{i,j} |K_{1}|Ψ_{((A^{+})^{−},g_{1}^{−},K_{1}^{−})}^{(\overline{(\overline{M}^{+})}^{−},Z,Z_{1})}(β_{J}, β_{i})δ^{i,j}Ψ_{((A^{+})^{+},g_{1}^{+},K_{1}^{+})}^{(\overline{(\overline{M}^{+})}^{+},Z_{1})}(β_{j})

= ∑_{i,j} |K_{1}|Ψ_{(A_{2}^{−},g_{2}^{−},K_{1}^{−})}^{(\overline{M}^{−},Z_{1})}(α_{1}^{−}, ⋯, α_{m}^{−}, β_{i})δ^{i,j}Ψ_{((A^{+})^{+},g_{1}^{+},K_{1}^{+})}^{(\overline{M}^{+},Z_{1})}(β_{j})
= Ψ_{C′}, for C′ = {(A^{+})^{+}, g_{1}^{+}, 0, K_{1}^{+}; A_{2}^{−}, g_{2}^{−}, m, K_{1}^{−}}

where g_{1}^{+} < g^{+}. In the second equality we use the gluing formula and the fact that A_{2}^{−} is the gluing class of A^{−} and (A^{+})^{−}, and the identifications \overline{M}^{+} ≅ \overline{\overline{M}^{+}}^{+} ≅ P(N_{C} ⊕ O) and the fact that the symplectic gluing of \overline{M}^{−} and (\overline{M^{+}})^{−} also recovers \overline{M}^{−}. Therefore, we may reduce the contribution of C to that of another component C′ = {(A^{+})^{+}, g_{1}^{+}, 0, K_{1}^{+}; A_{2}^{−}, g_{2}^{−}, m, K_{1}^{−}} with g_{1}^{+} < g^{+} by choosing a smaller symplectic cutting. By the finiteness of the genus g^{+}, after finite steps, we may reduce the component C to the case g^{+} = 0. Since we only have finite many gluing components with nonzero contribution, see [LR], so our claim follows from the argument of the case g^{+} = 0 and ν > 0.

So now it remains to show

ΨM̃ ; (p!(A),g)(p^{∗}α_{1}, …, p^{∗}α_{m}) = Ψ_{(A^{−},g,m)}^{(\overline{M}^{−},Z)}(α_{1}^{−}, …, α_{m}^{−}).

To prove this, we perform the symplectic cutting for M̃ around E. Therefore, we have

\overline{M̃}^{+} = P(N_{E} ⊕ O), \overline{M̃}^{−} ≅ M̃.

Now we use the gluing theorem to prove the contribution of stable J-holomorphic curves in M̃ which touch the exceptional divisor E to the GW-invariants of M̃ is zero. We consider the component

C = {p!(A)^{+}, g^{+}, m^{+}, K^{+}; p!(A)^{−}, g^{−}, m^{−}, K^{−}}.

Since α_{i}^{+} = 0, 1 ≤ i ≤ m, we have Ψ_{C} = 0 except

C = {p!(A)^{+}, g^{+}, 0, K^{+}; p!(A)^{−}, g^{−}, m, K^{−}}.

The same index calculation as in the proof of Theorem 1.5 of [H1] shows

IndD_{u^{+}} ≥ 2(n − 3)(l^{+} − g^{+}) + 2(2n − 3) ∑ k_{i} + 2ν
IndD_{u^{−}} ≤ 2C_{1}(A) + (2n − 6)(1 − l^{+} + g^{+} − g)
+(2n − 2)(ν − ∑ k_{i}) − 2(n − 2) ∑ k_{i} + 2m.

The same argument as in the first part of our proof shows that the contribution of J-holomorphic curves to the GW-invariant is nonzero only if it doesn’t touch the exceptional divisor E, i. e. C = {p!(A)^{−}, g, m}. Therefore, we have

ΨM̃ ; (p!(A),g)(p^{∗}α_{1}, …, p^{∗}α_{m}) = Ψ_{(p!(A)^{−},g,m)}^{(\overline{M̃}^{−},Z)}((p^{∗}α_{1})^{−}, …, (p^{∗}α_{m})^{−}).

The rest of the proof is the same as that of the proof of Theorem 1.2 of [H1]. So we omit it here. This proves Theorem 1.

Proof of Theorem 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 N_{δ}(S) of S which is symplectomorphic to the normal bundle N_{S}. We perform the symplectic cutting as in section 2.1 of [H1]. We obtain

\overline{M}^{+} = P(N_{S} ⊕ O), \overline{M}^{−} = M̃.

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, we first consider the contribution of each component to the GW-invariants. Therefore, we consider the component

C = {A^{+}, g^{+}, m^{+}, K^{+}; A^{−}, g^{−}, m^{−}, K^{−}}.

¿From Proposition 2.1, we have

IndD_{u^{+}} + IndD_{u^{−}} = 2(n − 1)ν + 2C_{1}(A) + 2(n − 3)(1 − g) + 2m.

We assume u^{±} : Σ^{±} ⟶ \overline{M}^{±} has l^{±} connected components u_{i}^{±} : Σ_{i}^{±} ⟶ \overline{M}^{±}, i = 1, …, l^{±}. Suppose Σ_{i}^{±} has arithemetic genus g_{i}^{±}, g^{±} = ∑ g_{i}^{±} with m^{±} marked points and m^{±} = ∑ m_{i}^{±}.

The same index calculation as in the proofs of Theorem 1.6 of [H1] and Theorem of [H2] shows

IndD_{u^{+}} = 2(n − 3)(l^{+} − g^{+}) + 2(n − 2) ∑ k_{i} + 2ν + 2m^{+}.

Therefore, we have

IndD_{u^{−}} = 2C_{1}(A) + 2(n − 3)(1 − l^{+} + g^{+} − g)
+2(n − 2)(ν − ∑ k_{i}) + 2m^{−}.

The same argument as in the proof of Theorem 1 shows that the contribution of the component C to the GW-invariant of M is nonzero only if C is the form

C = {A^{+}, g^{+}, 0, K^{+}; A^{−}, g^{−}, m, K^{−}}.

We also assume

∑ degα_{i} = 2C_{1}(A) + 2(n − 3)(1 − g) + 2m.

The same argument as in the proof of theorem 1 shows that there is a symplectic cutting such that Ψ_{C} = 0 except C = {A^{−}, g, m}. ¿From the gluing theorem, we have

Ψ_{(A,g)}^{M}(α_{1}, …, α_{m}) = Ψ_{(A^{−},g,m)}^{(\overline{M}^{−},Z)}(α_{1}^{−}, …, α_{m}^{−}).

Now it remains to prove

ΨM̃ ; (p!(A),g)(p^{∗}α_{1}, …, p^{∗}α_{m}) = Ψ_{(A^{−},g,m)}^{(\overline{M}^{−},Z)}(α_{1}^{−}, …, α_{m}^{−}).

To prove this, we perform the symplectic cutting for M̃ around E as in the proof of Theorem 1. Therefore, we have

\overline{M̃}^{+} = P(N_{E} ⊕ O), \overline{M̃}^{−} ≅ M̃.

We also use the gluing theorem to prove that the contribution of stable J-holomorphic curves in M̃ which touch the exceptional divisor E to the GWinvariant of M̃ is zero. We consider the component

C = {p!(A)^{+}, g^{+}, m^{+}, K^{+}; p!(A)^{−}, g^{−}, m^{−}, K^{−}}.

Since α_{i}^{+} = 0, 1 ≤ i ≤ m, we have Ψ_{C} = 0 except

C = {p!(A)^{+}, g^{+}, 0, K^{+}; p!(A)^{−}, g^{−}, m, K^{−}}.

The similar calculation to that in the proof of Theorem 1 shows

IndD_{u^{+}} = 2(n − 3)(l^{+} − g^{+}) + 2ν + 2(2n − 5) ∑ k_{i},
IndD_{u^{−}} = 2C_{1}(A) + 2(n − 3)(1 − l^{+} + g^{+} − g)
+2(n − 2)(ν − ∑ k_{i}) − 2(n − 3) ∑ k_{i} + 2m.

The same argument as in the proof of Theorem 1 shows

ΨM̃ ; (p!(A),g)(p^{∗}α_{1}, …, p^{∗}α_{m}) = Ψ_{(p!(A)^{−},g,m)}^{(\overline{M̃}^{−},Z)}((p^{∗}α_{1})^{−}, …, (p^{∗}α_{m})^{−}).

The rest of the proof is the same as that of the proof Theorem 1.2 of [H1]. So we omit it. This completes the proof of Theorem 2.

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Email address: stsjxhu@zsu.edu.cn