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**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.

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.) |

### Citations:

Zbl 0592.53025
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\textit{H. Hofer} et al., J. Geom. Anal. 7, No. 1, 149--159 (1997; Zbl 0911.53014)

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### References:

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[2] | Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1978), New York: Wiley, New York · Zbl 0408.14001 |

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[4] | Ivashkovich, S., and Shevchishin, V. Pseudoholomorphic curves and envelopes of meromorphy of two-spheres in CP^2, preprint, Essen/Bochum/Dusseldorf SFB237, January 1995. |

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[6] | McDuff, D., The local behaviour of holomorphic curves in almost-complex 4-manifolds, J. Diff. Geom., 34, 143-164 (1991) · Zbl 0736.53038 |

[7] | McDuff, D., and Salamon, D. Pseudo-holomorphic curves. Manuscript. · Zbl 1272.53002 |

[8] | McDuff, D., and Salamon, D.J-holomorphic curves and quantum cohomology.Amer. Math. Soc. Lect. Notes6 (1994). · Zbl 0809.53002 |

[9] | Sikorav, J. C.; Audin, M.; Lafontaine, J., Some properties of holomorphic curves in almost complex manifolds, Chapter V ofHolomorphic Curves in Symplectic Geometry (1994), Basel: Birkhäuser, Basel |

[10] | Vekua, N. I., Generalized Analytic Functions (1962), London: Pergamon, London · Zbl 0100.07603 |

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