Analysis Seminar

Speaker: Octavian, Mitrea (Western)

"A characterization of rationally convex immersions"

Time: 15:30

Room: MC 108

The notions of rational and polynomial convexity of compact subsets of the Euclidean complex space $\mathbb{C}^n$ plays a fundamental role in the general theory of approximation of continuous functions, uncovering deep connections to topology, Banach algebras, symplectic geometry and other areas of mathematics.

In 1995 Duval and Sibony proved that the rational convexity of any smooth, totally real submanifold of $\mathbb C^{n}$ is equivalent with the submanifold being isotropic with respect with some K\"ahler form defined on $\mathbb{C}^n$. In 2000, Gayet showed that if a totally real immersion in $\mathbb{C}^n$ of maximal real dimension with finitely many transverse self-intersections is Lagrangian with respect to some K\"ahler form, then that immersion is rationally convex.

In this series of two talks we prove a generalization of the above results. More specifically, we consider a more general class of totally real compact immersions, whose dimension does not have to be maximal, with finitely many self-intersecting points which are not necessarily transverse. We show that the rational convexity of such immersions is equivalent with the existence of a family of "degenerate" K\"ahler forms defined on $\mathbb{C}^n$ with respect to which the immersions are isotropic. By constructing a concrete example, we also show that our criterion is sharp, in the sense that there exist such immersions that are rationally convex but they are not isotropic with respect to any genuine K\"ahler form in $\mathbb{C}^n$.