On compact sets possessing $q$-convex functions

TL;DR

This paper characterizes when a compact set in a complex manifold admits a neighborhood with a $q$-convex function, linking it to the maximal $q$-pseudoconcave subset of the set.

Contribution

It provides a necessary and sufficient condition for the existence of $q$-convex functions near compact sets based on the $q$-nucleus and $q$-pseudoconcavity.

Findings

Existence of $q$-convex functions is equivalent to the emptiness of the $q$-nucleus.

The $q$-nucleus is characterized as the maximal $q$-pseudoconcave subset of the compact set.

The result offers a geometric criterion for $q$-convexity in complex manifolds.

Abstract

We show that there exists a $q$-convex function in a neighborhood of a compact set $K$ in a complex manifold $\mathcal{M}$ if and only if the $q$-nucleus of this compact set is empty. The latter can be characterized as the maximal $q$-pseudoconcave subset of $K$, i.e., a subset of $K$ containing all other compact $q$-pseudoconcave subsets in $K$.