##
**Symplectic hypersurfaces and transversality in Gromov-Witten theory.**
*(English)*
Zbl 1149.53052

Applying Donaldson’s construction of symplectic hypersurfaces [S. K. Donaldson, J. Differ. Geom. 44, No. 4, 666–705 (1996; Zbl 0883.53032)], transversality for holomorphic spheres in closed symplectic manifolds is proved using only standard tools from functional analysis. Moduli spaces of genus zero nodal curves constructed in this paper are smooth and their dimensions are explicitly computed. Then, following methods of D. McDuff and D. Salamon [\(J\)-holomorphic curves and symplectic topology, Colloquium Publications. American Mathematical Society 52. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1064.53051)], hereafter referred to as [1], a definition of genus zero Gromov-Witten invariants is given by using these results.

Before stating the main results of the paper, we give some definitions: Let \((X^{2n},\omega)\) be a closed symplectic manifold such that the de Rham class \([\omega]\) belongs to \(H^2(X,\mathbb{Z})\). Then, Donaldson’s construction asserts the existence of an approximate \(J\)-holomorphic hypersurface \(Y\subset X\) whose Poincaré dual in \(H^2(X;\mathbb{Z})\) equals \(D[\omega]\), if \(D\) is sufficiently large. Moreover, such \(Y\) is stable and unique up to isotopy. Here \(J\) is an \(\omega\)-compatible almost complex structure of \(X\), that is \(\omega(\cdot, J\cdot)\) is a Riemannian metric. Donaldson pair means the pair \((J, Y)\). Fixing a Donaldson pair \((J,Y)\), a space \({\mathcal J}_{\ell+1}(X, Y;J,\theta_1)\) of coherent almost complex structures on \(X\) depending smoothly on points in the Deligne-Mumford space \(\overline{{\mathcal M}}_{\ell+1}\) of stable genus zero curves with \(\ell+ 1\) marked points \(z_0,\dots, z_\ell\) and \(\theta_1\)-close to \(J\) in the \(C^0\)-norm is defined (§3. Definition of coherent and the Deligne-Mumford space are given in §3). \(K\in{\mathcal J}_{\ell+1}(X,Y; J,\theta_1)\) is thought to be a collection of almost complex structures \(K_{\mathbf z}\), parametrized by \({\mathbf z}\in\overline{{\mathcal M}}_{\ell+1}\). To \(K\) and a stable map \(({\mathbf z}, {\mathbf f})\) with \(\ell\) marked points, the \((0,1)\)-form \(\overline\partial_{K_{\text{st}({\mathbf z})}}{\mathbf f}\) is associated (§4). \(({\mathbf z}, {\mathbf f})\) is called \(K\)-holomorphic if \(\overline\partial_{K_{\text{st}({\mathbf z})}}{\mathbf f}= 0\).

Let \(K\in{\mathcal J}_{\ell+1}(X,Y; J,\theta_1)\). Then the map \(\pi_\ell: \overline{{\mathcal M}}_{k+\ell+1}\to \overline{{\mathcal M}}_{\ell+1}\) induces \(\pi^*_\ell K\in{\mathcal J}_{\ell+k+1}(X,Y;\) \(J,\theta_1)\). Using this, if \(A\subset H_2(X,\mathbb{Z})\) satisfies \(D\omega(A)=\ell\), the moduli space \({\mathcal M}_{k+\ell}(A,K; Y)\) of (smooth) \(K\)-holomorphic spheres in the class \(A\) with \(k+\ell\) marked points mapping the last \(\ell\) marked points to \(Y\), is defined. A \((k+\ell)\)-labelled tree \(T\) is said to be \(\ell\)-stable if it is stable after removing the first \(k\) marked points (§2). Let \(A_\alpha\) be homology classes corresponding to vertices \(\alpha\) of \(T\). The space of stable \(K\)-holomorphic spheres modeled over \(T\), representing \(A_\alpha\) and mapping the last \(\ell\) marked points to \(Y\) is denoted by \({\mathcal M}_T(\{A_\alpha\}, K; Y)\). Then the main results of the paper are

Theorem 1.1. If \(D\) is large and \(\ell= D\omega(A)\), then there exist nonempty sets \({\mathcal J}^{\text{reg}}_{\ell+1}(X, Y; J,\theta_1)\subset{\mathcal J}_{\ell+1}(X,Y; J,\theta_1)\) such that for suitable \(T\), the moduli space \({\mathcal M}_T(\{A_\alpha\}, K: Y)\), \(\sum A_\alpha= A\), is a smooth manifold and

\[ \dim_{\mathbb{R}}{\mathcal M}_T(\{A_\alpha\}, K; Y)= 2(n- 3+ k+ c_1(A)- e(T)). \]

Here \(c_1\) is the first Chern class of \(X\) and \(e(T)\) is the number of edges of \(T\).

Theorem 1.2. The evaluation map \(\text{ev}^k: M_{k+\ell}(A,K;Y)\to X^k\), \(k\geq 1\), at the first \(k\) points represents a pseudococycle \(\text{ev}^k(J,Y;K)\) of dimension \(2d= 2(n- 3+ k+ c_1(A))\).

Theorem 1.3. Up to rational cobordism, the rational pseudococycle \({1\over \ell!} \text{ev}^k(J,Y; K)\) in Theorem 1.2 does not depend on the choice of the auxiliarity data \((J,Y; K)\).

By Theorems 1.2 and 1.3, the Gromov-Witten invariant for arbitrary closed symplectic manifolds \((X,\omega)\) is defined as follows: Let \(\alpha_1,\dots, \alpha_k\) be nontorsion integral cohomology classes on \(X\) of total degree \(\sum^k_{i=1} \deg(\alpha_i)= 2d\). Let \(a\) be the Poincaré dual of \(\pi^*_1 \alpha_1\cup\cdots\cup \pi^*_k \alpha_k\in H^*(X^k)\). By Theorem 1.2, \(a\) is strongly transverse to \(\text{ev}^k(J,Y; K)\). Then, the Gromov-Witten invariant of holomorphic spheres passing through cycles dual to \(\alpha_1,\dots, \alpha_k\) is defined as the intersection number \[ \text{GW}_A(\alpha_1,\dots, \alpha_k)= {1\over\ell!} \text{ev}^k(J,Y; K)\cdot a\in\mathbb{Q}. \]

Theorems 1.3 and 1.2 show that this number is independent of the choices of \((J,Y; K)\).

The authors say that this definition of Gromov-Witten invariant coincides with that of McDuff-Salamon given in [1], if \((X,\omega)\) is semipositive, but the comparison with definitions involving abstract perturbation and virtual moduli cycles [K. Fukaya, and K. Ono, Topology 38, No. 5, 933–1048 (1999; Zbl 0946.53047), G. Liu and G. Tian, J. Differ. Geom. 49, No. 1, 1–74 (1998; Zbl 0917.58009)] lies beyond the scope of this paper. The authors also say that this paper is clearly related to the work of relative Gromov-Witten invariants of E.-N. Ionel and T. Parker, [Ann. Math. (2) 157, No. 1, 45–96 (2003; Zbl 1039.53101)], although relative Gromov-Witten invariants are not defined in this paper.

Sections 2–4 prepare necessary definitions and tools for the proof of transversality. The space \(\overline{{\mathcal M}}_T\) of all nodal curves of genus zero with \(k\) marked points modeled over a tree \(T\) is an open subset of the product of spheres. If \(T\) is stable, then the group \(G_T\) of isomorphisms fixing \(T\) acts on \(\overline{{\mathcal M}}_T\) freely. Let \({\mathcal M}_T\) be \(\overline{{\mathcal M}}_T/G_T\), then the moduli space of stable curves modelled over a \(k\)-labelled tree with one vertex is \({\mathcal M}_k= \overline{{\mathcal M}}_k/G\), for \(k\geq 3\), where \(\overline{{\mathcal M}}_k=\coprod_T{\mathcal M}_T\) (§2). Its compactification in the Gromov topology is the Deligne-Mumford space \(\overline{{\mathcal M}}_k\). Then, the notion of coherent map \(F:\overline{{\mathcal M}}_k\to Z\), \(Z\) a Banach space, and constructions of coherent maps are explained (the notion of Section 3 also studies almost complex structures on \(X\) using the topology induced by Floer’s \(C^\varepsilon\)-norm [A. Floer, Commun. Pure Appl. Math. 41, No. 6, 775–813 (1988; Zbl 0633.53058)]. Section 4 studies \(J\)-holomorphic maps and show

\[ \overline{{\mathcal M}}({\mathcal J}_{S^2})= \{(f, J)\in{\mathcal B}\times{\mathcal J}_{S^2} \|\overline\partial_J f= 0\},\quad{\mathcal B}= W^{m, p}(S^2, X), \]

is a manifold and the 1-point evaluation map \(\text{ev}^1:\overline{{\mathcal M}}({\mathcal J}_{S^2})\to X\) is a submersion. An ordal map \({\mathbf f}= \{f_\alpha\}_{\alpha\in T}\) represents a homology class \([{\mathbf f}]= \sum_{\alpha\in T}[f_\alpha]\in H_2(X;\mathbb{Z})\). Hence, the spaces \(\overline{{\mathcal M}}_T(\{A_\alpha\}, J)\) and \(\overline{{\mathcal M}}_T(A, J)\), \(A= \sum_\alpha A_\alpha\) are defined. Then, applying Gromov’s compactness theorem to these spaces, an open subset \(\overline{{\mathcal M}}^*_T(\{A_\alpha\},J)\) of \(\overline{{\mathcal M}}_T(\{A_\alpha\}, J)\), such that

\[ \overline{{\mathcal M}}_T^*(\{A_\alpha\},{\mathcal J}_{|I|+ 1}(V))= \bigcup_{J\in{\mathcal J}_{|I|+1}(V)} \overline{{\mathcal M}}_T^*(\{A_\alpha\}, J) \]

becomes a Banach manifold, is extracted. Here, \(V\) is an open set of \(X\). It is also shown \(\text{ev}^R: \overline{{M}}_T^*(\{A_\alpha\},{\mathcal J}_{|I|+ 1}(V))\to X^R\) is a submersion. As corollaries, the existence of \(J\in{\mathcal J}_{|\ell|+1}(V)\) such that \({\mathcal M}^*_T(\{A_\alpha\}, J;Z)\) becomes a smooth manifold together with computation of its dimension and factorization \[ \text{ev}^k:{\mathcal M}^*_T(\{A_\alpha\}, J;Z)\to {\mathcal M}^*_{\pi_R(T)}(\{A_\alpha\}, J: Z)\to X^k \]

are obtained. In Section 6, moduli spaces of \(J\)-holomorphic spheres with prescribed orders of tangency to complex submanifolds are defined. Then, their regularities are proved and dimensions are computed. Section 7 defines intersection numbers \(\iota({\mathbf f},Y; z_i)\) of \({\mathbf f}\) and a complex hypersurface \(Y\) of \(X\) at a marked point \(z_i\). These results are connected to the geometry of \((X,\omega)\) via Donaldson’s construction. After these preparations, the theorems 1.1, 1.2 and 1.3 are proved.

The authors say that the method in this paper is applicable to prove transversality for genus zero curves with boundary on a Lagrangian submanifold and genus zero holomorphic curves with punctures asymptotic to closed Reeb orbits are possible, although the moduli spaces of these cases have codimension 1 boundary. These extensions may be treated in subsequent papers.

Before stating the main results of the paper, we give some definitions: Let \((X^{2n},\omega)\) be a closed symplectic manifold such that the de Rham class \([\omega]\) belongs to \(H^2(X,\mathbb{Z})\). Then, Donaldson’s construction asserts the existence of an approximate \(J\)-holomorphic hypersurface \(Y\subset X\) whose Poincaré dual in \(H^2(X;\mathbb{Z})\) equals \(D[\omega]\), if \(D\) is sufficiently large. Moreover, such \(Y\) is stable and unique up to isotopy. Here \(J\) is an \(\omega\)-compatible almost complex structure of \(X\), that is \(\omega(\cdot, J\cdot)\) is a Riemannian metric. Donaldson pair means the pair \((J, Y)\). Fixing a Donaldson pair \((J,Y)\), a space \({\mathcal J}_{\ell+1}(X, Y;J,\theta_1)\) of coherent almost complex structures on \(X\) depending smoothly on points in the Deligne-Mumford space \(\overline{{\mathcal M}}_{\ell+1}\) of stable genus zero curves with \(\ell+ 1\) marked points \(z_0,\dots, z_\ell\) and \(\theta_1\)-close to \(J\) in the \(C^0\)-norm is defined (§3. Definition of coherent and the Deligne-Mumford space are given in §3). \(K\in{\mathcal J}_{\ell+1}(X,Y; J,\theta_1)\) is thought to be a collection of almost complex structures \(K_{\mathbf z}\), parametrized by \({\mathbf z}\in\overline{{\mathcal M}}_{\ell+1}\). To \(K\) and a stable map \(({\mathbf z}, {\mathbf f})\) with \(\ell\) marked points, the \((0,1)\)-form \(\overline\partial_{K_{\text{st}({\mathbf z})}}{\mathbf f}\) is associated (§4). \(({\mathbf z}, {\mathbf f})\) is called \(K\)-holomorphic if \(\overline\partial_{K_{\text{st}({\mathbf z})}}{\mathbf f}= 0\).

Let \(K\in{\mathcal J}_{\ell+1}(X,Y; J,\theta_1)\). Then the map \(\pi_\ell: \overline{{\mathcal M}}_{k+\ell+1}\to \overline{{\mathcal M}}_{\ell+1}\) induces \(\pi^*_\ell K\in{\mathcal J}_{\ell+k+1}(X,Y;\) \(J,\theta_1)\). Using this, if \(A\subset H_2(X,\mathbb{Z})\) satisfies \(D\omega(A)=\ell\), the moduli space \({\mathcal M}_{k+\ell}(A,K; Y)\) of (smooth) \(K\)-holomorphic spheres in the class \(A\) with \(k+\ell\) marked points mapping the last \(\ell\) marked points to \(Y\), is defined. A \((k+\ell)\)-labelled tree \(T\) is said to be \(\ell\)-stable if it is stable after removing the first \(k\) marked points (§2). Let \(A_\alpha\) be homology classes corresponding to vertices \(\alpha\) of \(T\). The space of stable \(K\)-holomorphic spheres modeled over \(T\), representing \(A_\alpha\) and mapping the last \(\ell\) marked points to \(Y\) is denoted by \({\mathcal M}_T(\{A_\alpha\}, K; Y)\). Then the main results of the paper are

Theorem 1.1. If \(D\) is large and \(\ell= D\omega(A)\), then there exist nonempty sets \({\mathcal J}^{\text{reg}}_{\ell+1}(X, Y; J,\theta_1)\subset{\mathcal J}_{\ell+1}(X,Y; J,\theta_1)\) such that for suitable \(T\), the moduli space \({\mathcal M}_T(\{A_\alpha\}, K: Y)\), \(\sum A_\alpha= A\), is a smooth manifold and

\[ \dim_{\mathbb{R}}{\mathcal M}_T(\{A_\alpha\}, K; Y)= 2(n- 3+ k+ c_1(A)- e(T)). \]

Here \(c_1\) is the first Chern class of \(X\) and \(e(T)\) is the number of edges of \(T\).

Theorem 1.2. The evaluation map \(\text{ev}^k: M_{k+\ell}(A,K;Y)\to X^k\), \(k\geq 1\), at the first \(k\) points represents a pseudococycle \(\text{ev}^k(J,Y;K)\) of dimension \(2d= 2(n- 3+ k+ c_1(A))\).

Theorem 1.3. Up to rational cobordism, the rational pseudococycle \({1\over \ell!} \text{ev}^k(J,Y; K)\) in Theorem 1.2 does not depend on the choice of the auxiliarity data \((J,Y; K)\).

By Theorems 1.2 and 1.3, the Gromov-Witten invariant for arbitrary closed symplectic manifolds \((X,\omega)\) is defined as follows: Let \(\alpha_1,\dots, \alpha_k\) be nontorsion integral cohomology classes on \(X\) of total degree \(\sum^k_{i=1} \deg(\alpha_i)= 2d\). Let \(a\) be the Poincaré dual of \(\pi^*_1 \alpha_1\cup\cdots\cup \pi^*_k \alpha_k\in H^*(X^k)\). By Theorem 1.2, \(a\) is strongly transverse to \(\text{ev}^k(J,Y; K)\). Then, the Gromov-Witten invariant of holomorphic spheres passing through cycles dual to \(\alpha_1,\dots, \alpha_k\) is defined as the intersection number \[ \text{GW}_A(\alpha_1,\dots, \alpha_k)= {1\over\ell!} \text{ev}^k(J,Y; K)\cdot a\in\mathbb{Q}. \]

Theorems 1.3 and 1.2 show that this number is independent of the choices of \((J,Y; K)\).

The authors say that this definition of Gromov-Witten invariant coincides with that of McDuff-Salamon given in [1], if \((X,\omega)\) is semipositive, but the comparison with definitions involving abstract perturbation and virtual moduli cycles [K. Fukaya, and K. Ono, Topology 38, No. 5, 933–1048 (1999; Zbl 0946.53047), G. Liu and G. Tian, J. Differ. Geom. 49, No. 1, 1–74 (1998; Zbl 0917.58009)] lies beyond the scope of this paper. The authors also say that this paper is clearly related to the work of relative Gromov-Witten invariants of E.-N. Ionel and T. Parker, [Ann. Math. (2) 157, No. 1, 45–96 (2003; Zbl 1039.53101)], although relative Gromov-Witten invariants are not defined in this paper.

Sections 2–4 prepare necessary definitions and tools for the proof of transversality. The space \(\overline{{\mathcal M}}_T\) of all nodal curves of genus zero with \(k\) marked points modeled over a tree \(T\) is an open subset of the product of spheres. If \(T\) is stable, then the group \(G_T\) of isomorphisms fixing \(T\) acts on \(\overline{{\mathcal M}}_T\) freely. Let \({\mathcal M}_T\) be \(\overline{{\mathcal M}}_T/G_T\), then the moduli space of stable curves modelled over a \(k\)-labelled tree with one vertex is \({\mathcal M}_k= \overline{{\mathcal M}}_k/G\), for \(k\geq 3\), where \(\overline{{\mathcal M}}_k=\coprod_T{\mathcal M}_T\) (§2). Its compactification in the Gromov topology is the Deligne-Mumford space \(\overline{{\mathcal M}}_k\). Then, the notion of coherent map \(F:\overline{{\mathcal M}}_k\to Z\), \(Z\) a Banach space, and constructions of coherent maps are explained (the notion of Section 3 also studies almost complex structures on \(X\) using the topology induced by Floer’s \(C^\varepsilon\)-norm [A. Floer, Commun. Pure Appl. Math. 41, No. 6, 775–813 (1988; Zbl 0633.53058)]. Section 4 studies \(J\)-holomorphic maps and show

\[ \overline{{\mathcal M}}({\mathcal J}_{S^2})= \{(f, J)\in{\mathcal B}\times{\mathcal J}_{S^2} \|\overline\partial_J f= 0\},\quad{\mathcal B}= W^{m, p}(S^2, X), \]

is a manifold and the 1-point evaluation map \(\text{ev}^1:\overline{{\mathcal M}}({\mathcal J}_{S^2})\to X\) is a submersion. An ordal map \({\mathbf f}= \{f_\alpha\}_{\alpha\in T}\) represents a homology class \([{\mathbf f}]= \sum_{\alpha\in T}[f_\alpha]\in H_2(X;\mathbb{Z})\). Hence, the spaces \(\overline{{\mathcal M}}_T(\{A_\alpha\}, J)\) and \(\overline{{\mathcal M}}_T(A, J)\), \(A= \sum_\alpha A_\alpha\) are defined. Then, applying Gromov’s compactness theorem to these spaces, an open subset \(\overline{{\mathcal M}}^*_T(\{A_\alpha\},J)\) of \(\overline{{\mathcal M}}_T(\{A_\alpha\}, J)\), such that

\[ \overline{{\mathcal M}}_T^*(\{A_\alpha\},{\mathcal J}_{|I|+ 1}(V))= \bigcup_{J\in{\mathcal J}_{|I|+1}(V)} \overline{{\mathcal M}}_T^*(\{A_\alpha\}, J) \]

becomes a Banach manifold, is extracted. Here, \(V\) is an open set of \(X\). It is also shown \(\text{ev}^R: \overline{{M}}_T^*(\{A_\alpha\},{\mathcal J}_{|I|+ 1}(V))\to X^R\) is a submersion. As corollaries, the existence of \(J\in{\mathcal J}_{|\ell|+1}(V)\) such that \({\mathcal M}^*_T(\{A_\alpha\}, J;Z)\) becomes a smooth manifold together with computation of its dimension and factorization \[ \text{ev}^k:{\mathcal M}^*_T(\{A_\alpha\}, J;Z)\to {\mathcal M}^*_{\pi_R(T)}(\{A_\alpha\}, J: Z)\to X^k \]

are obtained. In Section 6, moduli spaces of \(J\)-holomorphic spheres with prescribed orders of tangency to complex submanifolds are defined. Then, their regularities are proved and dimensions are computed. Section 7 defines intersection numbers \(\iota({\mathbf f},Y; z_i)\) of \({\mathbf f}\) and a complex hypersurface \(Y\) of \(X\) at a marked point \(z_i\). These results are connected to the geometry of \((X,\omega)\) via Donaldson’s construction. After these preparations, the theorems 1.1, 1.2 and 1.3 are proved.

The authors say that the method in this paper is applicable to prove transversality for genus zero curves with boundary on a Lagrangian submanifold and genus zero holomorphic curves with punctures asymptotic to closed Reeb orbits are possible, although the moduli spaces of these cases have codimension 1 boundary. These extensions may be treated in subsequent papers.

Reviewer: Akira Asada (Takarazuka)

### MSC:

53D35 | Global theory of symplectic and contact manifolds |

53C40 | Global submanifolds |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |