A description of characteristic classes of real submanifolds in complex manifolds via RC-singularities

Let $X$ be a complex manifold, $M\subset X$ a (closed, orient) real submanifold, and $S\subset M$ the set of RC-singular points of $M$. We study the connection between the global topological characteristics of $S$ and the topology of $M$ and $X$. For the case of discrete $S$ we introduce a notion of an isolate RC-singular point and obtain a formula expressing the sun of indices over $S$ in terms of the Chern classes of $X$ and the Pontryagin classes of $M$ and of the normal bundle to $M$ in $X$ (Theorem 1). In the general case we express the Poincare dual to $S$ (Theorem 2) and the Poincare duals to some cycles carried by subsets of $S$ (Theorem 3) in a similar way.