Polynomial convexity of $\bar{\partial }$-flat perturbations of totally real sets

Abstract

We show that if $X$ is a totally real $d$-dimensional manifold attached to a polynomially convex compact set $K$ in ${\mathbb{C}}^n$, $d<n$, then there are arbitrarily small perturbations $X^{\prime }$ of $X$ such that $K\cup X^{\prime }$ is polynomially convex. The perturbations are induced by diffeomorphisms of ${\mathbb{C}}^n$ fixing $K$, which are $\bar{\partial }$-flat on $K\cup X$, and which are arbitrarily $C^k$-close to the identity.