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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
14H10 Families, moduli of curves (algebraic)
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