Complex dimensions of real manifolds, attached analytic discs and parametric argument principle

Details Complex dimensions of real manifolds, attached analytic discs and parametric argument principle Complex dimensions of real manifolds, attached analytic discs and parametric argument principle

2007-04-23

arxiv.org/abs/0704.2871v2

Mathematics

Complex Variables

Scientific paper

Let $\Omega$ be a smooth real analytic submanifold of a complex manifold $X$. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold $\Omega$ admits, 2) lower bounds for dimensions of complex tangent spaces of $\Omega$, 3) a generalization of the argument principle for smooth families of holomorphic mappings from the standard complex disc to $X$. In particular, we obtain characterization of complex manifolds and their boundaries in terms of attached analytic discs. The special case when $\Omega$ is the graph, leads to new characterizations of holomorphic and $CR$ functions, and in particular, to solutions of some open problems about such functions.

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