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 $T_pM \subset T_p\mathbb{C}^n$ 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 $T_pM$ 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.