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

TL;DR

This paper explores the relationship between attached analytic discs, complex tangent spaces, and an argument principle generalization, providing new characterizations of complex manifolds and solutions to open problems in CR functions.

Contribution

It establishes a novel link between topological properties of analytic discs, tangent space dimensions, and a generalized argument principle for complex manifolds.

Findings

Characterization of complex manifolds via attached analytic discs

Lower bounds for complex tangent space dimensions

New solutions to open problems in CR functions

Abstract

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.