## Courbes pseudo-holomorphes équisingulières en dimension 4. (Equisingular pseudo-holomorphic curves in 4 dimensonal almost complex manifolds).(French)Zbl 0976.53037

Author’s abstract: Let $${\mathcal M}$$ denote the space of all pseudo-holomorphic curves of given genus and homology in an almost complex manifold $$(V,J)$$, and let $${\mathfrak S}$$ be a set of singular points of a curve $$C\in{\mathcal M}$$ (or more generally, a set of “conditions” on $$C$$). We give a numerical condition on $$C$$ and $${\mathfrak S}$$ under which the space $${\mathcal M}_{{\mathfrak S}}$$ of all curves having “the same” singularities as $$C$$ near each point of $${\mathfrak S}$$ is a submanifold of $${\mathcal M}$$ (in a neighborhood of $$C$$). As an application, we study the sets of pseudo-holomorphic lines in $$\mathbb{C}\mathbb{P}^2$$: we prove in particular, that any generic set, i.e., having only double points, is isotopic to a set of standard lines.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58D27 Moduli problems for differential geometric structures 32Q65 Pseudoholomorphic curves 53D05 Symplectic manifolds (general theory) 14H10 Families, moduli of curves (algebraic)
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### References:

 [1] AUROUX (D.) . - Symplectic 4-manifolds as Branched Coverings of CP2 , preprint (auroux\?math. polytechnique.fr). [2] BARRAUD (J.-F.) . - Sphères symplectiques à points doubles ordinaires positifs dans CP2 , C.R. Acad. Sciences Paris, t. 327, Série I, 1998 , p. 661-668. MR 99m:58035 | Zbl 01223396 · Zbl 1007.58007 [3] GROMOV (M.) . - Pseudo holomorphic curves in symplectic manifolds , Inventiones Math., t. 82, 1985 , p. 307-347. MR 87j:53053 | Zbl 0592.53025 · Zbl 0592.53025 [4] HOFER (H.) , LIZAN (V.) , SIKORAV (J.-C.) . - On genericity of holomorphic curves in 4 dimensional almost complex manifolds , J. Geometric Anal., t. 7, 1997 , p. 149-159. MR 2000d:32045 | Zbl 0911.53014 · Zbl 0911.53014 [5] IVASHKOVICH (S.) , SHEVCHISHIN (V.) . - Structure of the moduli space in a neighborhood of a cusp curve and meromorphic hulls , Invent. Math., t. 136, 1999 , p. 571-602. MR 2001d:32035 | Zbl 0930.32017 · Zbl 0930.32017 [6] LIU (A.) . - Some new applications of general wall-crossing formula, Gompf’s conjecture and its applications , Math. Research. Lett., t. 3, n^\circ 5, 1996 , p. 569-585. MR 97k:57038 | Zbl 0872.57025 · Zbl 0872.57025 [7] MCDUFF (D.) . - The local behaviour of holomorphic curves in almost complex 4-manifolds , J. Differential Geom., t. 34, 1991 , p. 143-164. MR 93e:53050 | Zbl 0736.53038 · Zbl 0736.53038 [8] MCDUFF (D.) . - Examples of symplectic structures , Inventiones Math., t. 89, 1987 , p. 13-36. MR 88m:58061 | Zbl 0625.53040 · Zbl 0625.53040 [9] MCDUFF (D.) , SALAMON (D.) . - J-holomorphic curves and quantum cohomology. , Amer. Math. Soc. Univ. Lect. Notes, t. 6, 1994 . MR 95g:58026 | Zbl 0809.53002 · Zbl 0809.53002 [10] MCDUFF (D.) et D. SALAMON . - A survey of symplectic 4-manifolds with b$$^{+}$$ = 1 , Turk. J. Math., t. 20, n^\circ 1, 1996 , p. 47-60. MR 97e:57028 | Zbl 0870.57023 · Zbl 0870.57023 [11] SIKORAV (J.-C.) . - Local properties of J curves . - Chapitre V de Holomorphic curves in symplectic geometry, M. Audin et J. Lafontaine éd., Progress in Math., Birkhäuser, Basel, 1994 . [12] SIKORAV (J.-C.) . - Singularities of J-holomorphic curves , Math. Zeit., t. 226, 1997 , p. 359-373. MR 98k:58060 | Zbl 0886.30032 · Zbl 0886.30032
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