## 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 $$\text{pr}_{\mathcal J}$$.

### MSC:

 32Q55 Topological aspects of complex manifolds 32Q65 Pseudoholomorphic curves

### Keywords:

envelope of meromorphy; pseudo-holomorphic curve
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