## On genericity for holomorphic curves in four-dimensional almost-complex manifolds.(English)Zbl 0911.53014

Let $$(V,J)$$ be an almost complex manifold of dimension four and $$C_0\subset (V,J)$$ a pseudo-holomorphic curve without boundary. M. Gromov [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] showed that the equation of these curves is elliptic and has a Fredholm index. He also proved that, if the Chern class $$c_1(C_0)\geq 1$$, then the space of pseudo-holomorphic curves near $$C_0$$ is a manifold of dimension equal the index. The purpose of this note is to give a proof of this theorem as well as corresponding one for a curve $$C_0$$ with boundary.
Let $$C_0\subset V$$ be a regular holomorphic curve with boundary $$\partial C_0$$ in a totally real surface $$W\subset V$$. In this paper, it is shown that, if the ambient Maslov number $$\mu (C_0,\partial C_0)\geq 1$$, then the space of pseudo-holomorphic curves $$(C,\partial C)\subset (V,W)$$ near $$(C_0,\partial C_0)$$ is a manifold of dimension equal to the index.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 35J60 Nonlinear elliptic equations 47A53 (Semi-) Fredholm operators; index theories 58J05 Elliptic equations on manifolds, general theory 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, $$p$$-analytic, etc.)

Zbl 0592.53025
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### References:

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