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.


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
Full Text: DOI


[1] Gromov, M., Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82, 307-347 (1985) · Zbl 0592.53025
[2] Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1978), New York: Wiley, New York · Zbl 0408.14001
[3] Hofer, H., Pseudo-holomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114, 515-563 (1993) · Zbl 0797.58023
[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.
[5] McDuff, D., Rational and ruled symplectic manifolds, J. Am. Math. Soc., 3, 679-711 (1990) · Zbl 0723.53019
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.