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**Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls.**
*(English)*
Zbl 0930.32017

Let us call a Kähler surface \((X,\omega)\) strongly minimal if for any almost-complex structure \(J\) on \(X\) tamed by \(\omega\), any \(J\)-holomorphic sphere \(C\subset X\) satisfies \([C]^2\geq 0\). For example \(CP^2\) and \(\mathbb{C}\mathbb{P}^1 \times\mathbb{C} \mathbb{P}^1\) are strongly minimal in this sense.

Consider an \(\omega\)-symplectic immersion \(u:S^1\to X\) of a standard two-dimensional sphere into \(X\), and suppose that \(M:= u(S^2)\) has only double and positive self-intersections. Denote by \(\delta\) the number of double points of our \(M\). \(c_1(X)\) will denote the first Chern class of \(X\) and \(c_1(X)[M]\) its value on \(M\).

Theorem 1. Under the conditions as above suppose that \(c_1(X) [M]- \delta= p\geq 1\). Then the envelope of meromorphy \((\widehat U,\pi)\) of any neighborhood \(U\) of \(M\) is either unbounded (i.e. \(\pi (\widehat U)\) is non-compact in \(X)\), or \(\widehat U\) contains a one-dimensional analytic set \(C\) such that

(1) \(\pi(C)= \cup^N_{k=1} C_k\) is a union of rational curves;

(2) \(\sum^N_{k=1} (c_1(\widehat U) [C_k]- \delta_k- \kappa_k)) \geq p\).

Here \(c_1 (\widehat U)= \pi^*c_1(X)\) is the first Chern class of \(\widehat U\), \(\delta_k\) the sum of local intersection numbers of \(C_k\), and \(\kappa_k\) – the sum of Milnor numbers of the cusps of \(C_k\).

Let \((X,J)\) be an almost complex manifold of complex dimension \(n\). An almost complex structure is always supposed to have smoothness of class \(C^1\). In what follows \((S,J_S)\) denotes a Riemann surface with complex structure \(J_S\).

It can be shown that for a pseudo-holomorphic curve \(u:(S,J_S) \to(X, J_0)\) the pull-back bundle \(E:= u^*TX\) possesses a natural holomorphic structure (the corresponding sheaf of holomorphic sections will be denoted as \({\mathcal O}(E))\), such that the differential \(du:TS\to E\) is a holomorphic homomorphism. This allows us to define the order of vanishing of the differential \(du\) at point \(s\in S\). We denote this number by \(\mathbf{ord}_sdu\).

This also gives the following short exact sequence: \[ 0\to {\mathcal O}(TS) @>du>> {\mathcal O}(E) @>\text{pr}>> {\mathcal O}(N_0) \oplus {\mathcal N}_1 \to 0, \] where \({\mathcal O}(N_0)\) denotes a free part of the quotient \({\mathcal O}(E)/du ({\mathcal O}(TS))\) and \({\mathcal N}_1\) is supported on a finite set of cusps of \(u\) (i.e. points of vanishing of \(du)\).

On the Sobolev space \(L^{1,p} (S,N_0)\) of \(L^{1,p}\)-smooth sections of the bundle \(N_0\) the natural Gromov operator \(D_N:L^{1,p} (S,N_0)\to L^p(S, \Lambda^{0,1} S \otimes N_0)\) is defined. Put \(H^0_D(S,N_0): =\text{Ker} D_N\) and \(H^1_D (S, N_0): =\text{Coker} D_N\).

The following Theorem is the second main result of this paper.

Theorem 2. Let \(u:(S,J_S) \to(X,J_0)\) be a non-constant pseudo-holomorphic map, such that \(H^1_D (S,N_0)=0\). Then:

i) in a neighbourhood of \(M:=u(S)\) the moduli space of nonparametrized \(J_0\)-holomorphic curves \({\mathcal M}_{[\gamma], g,J_0}\) is a manifold the tangent space of which is \(T_M {\mathcal M}_{[\gamma], g,J_0}= H^0_D (S,N_0) \oplus H^0(S, {\mathcal N}_1)\);

ii) further, there is a neighborhood \(V\ni J\) in the Banach manifold \({\mathcal I}\) of \(C^1\)-smooth almost-complex structures on \(X\) and a neighborhood \(W\) of \(M\) in \({\mathcal M}_{[\gamma],g,V}: =\cup_{J\in V} {\mathcal M}_{[\gamma], g,J}\) such, that the natural projection \(\text{pr}_J: W\to V\) is regular and with smooth fibers, in particular, for any \(M'\in W\) the conclusion i) is valid;

iii) if \(\dim_\mathbb{R} X=4\) and \(c_1(E)[M]>\sum_{s\in S}\text{ord}_sdu\), then \(H^1_D (S, N_0)=0\) and so the conclusions i) and ii) hold.

Remark. When \(X\) is a complex surface and \(N_0\) is a line bundle the sufficient condition for surjectivity of \(D_N\) is \(c_1(N_0) >2g-2\), where \(g\) is the genus of \(S\). So, if \(c_1(E)> \sum_{s\in S} \text{ord}_sdu\), one has the surjectivity of projection \(\text{pr}_{\mathcal J}\).

Consider an \(\omega\)-symplectic immersion \(u:S^1\to X\) of a standard two-dimensional sphere into \(X\), and suppose that \(M:= u(S^2)\) has only double and positive self-intersections. Denote by \(\delta\) the number of double points of our \(M\). \(c_1(X)\) will denote the first Chern class of \(X\) and \(c_1(X)[M]\) its value on \(M\).

Theorem 1. Under the conditions as above suppose that \(c_1(X) [M]- \delta= p\geq 1\). Then the envelope of meromorphy \((\widehat U,\pi)\) of any neighborhood \(U\) of \(M\) is either unbounded (i.e. \(\pi (\widehat U)\) is non-compact in \(X)\), or \(\widehat U\) contains a one-dimensional analytic set \(C\) such that

(1) \(\pi(C)= \cup^N_{k=1} C_k\) is a union of rational curves;

(2) \(\sum^N_{k=1} (c_1(\widehat U) [C_k]- \delta_k- \kappa_k)) \geq p\).

Here \(c_1 (\widehat U)= \pi^*c_1(X)\) is the first Chern class of \(\widehat U\), \(\delta_k\) the sum of local intersection numbers of \(C_k\), and \(\kappa_k\) – the sum of Milnor numbers of the cusps of \(C_k\).

Let \((X,J)\) be an almost complex manifold of complex dimension \(n\). An almost complex structure is always supposed to have smoothness of class \(C^1\). In what follows \((S,J_S)\) denotes a Riemann surface with complex structure \(J_S\).

It can be shown that for a pseudo-holomorphic curve \(u:(S,J_S) \to(X, J_0)\) the pull-back bundle \(E:= u^*TX\) possesses a natural holomorphic structure (the corresponding sheaf of holomorphic sections will be denoted as \({\mathcal O}(E))\), such that the differential \(du:TS\to E\) is a holomorphic homomorphism. This allows us to define the order of vanishing of the differential \(du\) at point \(s\in S\). We denote this number by \(\mathbf{ord}_sdu\).

This also gives the following short exact sequence: \[ 0\to {\mathcal O}(TS) @>du>> {\mathcal O}(E) @>\text{pr}>> {\mathcal O}(N_0) \oplus {\mathcal N}_1 \to 0, \] where \({\mathcal O}(N_0)\) denotes a free part of the quotient \({\mathcal O}(E)/du ({\mathcal O}(TS))\) and \({\mathcal N}_1\) is supported on a finite set of cusps of \(u\) (i.e. points of vanishing of \(du)\).

On the Sobolev space \(L^{1,p} (S,N_0)\) of \(L^{1,p}\)-smooth sections of the bundle \(N_0\) the natural Gromov operator \(D_N:L^{1,p} (S,N_0)\to L^p(S, \Lambda^{0,1} S \otimes N_0)\) is defined. Put \(H^0_D(S,N_0): =\text{Ker} D_N\) and \(H^1_D (S, N_0): =\text{Coker} D_N\).

The following Theorem is the second main result of this paper.

Theorem 2. Let \(u:(S,J_S) \to(X,J_0)\) be a non-constant pseudo-holomorphic map, such that \(H^1_D (S,N_0)=0\). Then:

i) in a neighbourhood of \(M:=u(S)\) the moduli space of nonparametrized \(J_0\)-holomorphic curves \({\mathcal M}_{[\gamma], g,J_0}\) is a manifold the tangent space of which is \(T_M {\mathcal M}_{[\gamma], g,J_0}= H^0_D (S,N_0) \oplus H^0(S, {\mathcal N}_1)\);

ii) further, there is a neighborhood \(V\ni J\) in the Banach manifold \({\mathcal I}\) of \(C^1\)-smooth almost-complex structures on \(X\) and a neighborhood \(W\) of \(M\) in \({\mathcal M}_{[\gamma],g,V}: =\cup_{J\in V} {\mathcal M}_{[\gamma], g,J}\) such, that the natural projection \(\text{pr}_J: W\to V\) is regular and with smooth fibers, in particular, for any \(M'\in W\) the conclusion i) is valid;

iii) if \(\dim_\mathbb{R} X=4\) and \(c_1(E)[M]>\sum_{s\in S}\text{ord}_sdu\), then \(H^1_D (S, N_0)=0\) and so the conclusions i) and ii) hold.

Remark. When \(X\) is a complex surface and \(N_0\) is a line bundle the sufficient condition for surjectivity of \(D_N\) is \(c_1(N_0) >2g-2\), where \(g\) is the genus of \(S\). So, if \(c_1(E)> \sum_{s\in S} \text{ord}_sdu\), one has the surjectivity of projection \(\text{pr}_{\mathcal J}\).

Reviewer: S.Ivashkovich (Villeneuve d’Ascq)