Local Polynomial Convexity at Hyperbolic CR-singularities in $M\subset {\mathbb{C}}^n$

Abstract

Let $M$ be a smooth manifold of dimension $n$ embedded in ${\mathbb{C}}^n$. If $T_{p}M\subset T_p{\mathbb{C}}^n$ is a totally real subspace for $p\in M$, then $M$ is locally polynomially convex at $p$. For a generic embedding $M$, we are interested in assessing polynomial convexity of $M$ at a CR-singularity, i.e., at a point $p\in M$ where $T_{p}M$ is not totally real. An order one CR-singularity in $M$ can be broadly classified as elliptic and hyperbolic. It is known that elliptic points give obstruction to polynomial convexity. In the case $n=2$, $M^2\subset {\mathbb{C}}^2$ is locally polynomially convex at a hyperbolic complex point. We investigate local polynomial convexity of $M^n\subset {\mathbb{C}}^n$ at hyperbolic points in higher dimension.

24 pages