A Characterization of Rationally Convex Immersions

Let $S$ be a smooth, totally real, compact immersion in ${\mathbb{C}}^n$ of real dimension $m\le n$, 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 ${\mathbb{C}}^n$.