A Characterization of Rationally Convex Immersions

Abstract

Let $S$ be a smooth, totally real, compact immersion in $\mathbb{C}^n$ of real dimension $m \leq n$, which is locally polynomially convex and it has finitely many points where it self-intersects finitely many times, transversely or non-transversely. We prove that $S$ is rationally convex if and only if it is isotropic with respect to a "degenerate" K\"ahler form in $\mathbb{C}^n$.Comment: In this second version of the paper, we strengthen the statement of the main theorem, address some typos that the first version contains and enhance the clarity of some parts of the proof of the main resul