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

TL;DR

This paper investigates the local polynomial convexity of smooth manifolds in complex space at hyperbolic CR-singularities, extending known results from two dimensions to higher dimensions.

Contribution

It extends the understanding of polynomial convexity at hyperbolic CR-singularities from complex dimension two to higher dimensions.

Findings

Hyperbolic points do not obstruct polynomial convexity in higher dimensions.

The paper characterizes conditions under which local polynomial convexity holds at hyperbolic CR-singularities.

Results generalize known two-dimensional cases to $n$-dimensional manifolds.

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.