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

Let $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ admits, 2) lower bounds for dimensions of complex tangent spaces of $\mathrm{\Omega }$, 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 $\mathrm{\Omega }$ is the graph, leads to new characterizations of holomorphic and $CR$ functions, and in particular, to solutions of some open problems about such functions.