A Characterization of Rationally Convex Immersions

TL;DR

This paper characterizes when a smooth, totally real, compact immersion in complex space is rationally convex, linking this property to isotropy with respect to a specific degenerate Kähler form.

Contribution

It establishes a necessary and sufficient condition for rational convexity of certain immersions based on isotropy with a degenerate Kähler form.

Findings

Rational convexity is equivalent to isotropy under a degenerate Kähler form.

The characterization applies to immersions with finitely many self-intersection points.

Provides a geometric criterion for rational convexity in complex Euclidean spaces.

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 $\mathbb{C}^n$.