Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces

TL;DR

The paper proves the existence of strongly q-convex exhaustion functions near projective subvarieties in complex spaces under certain covering space conditions, advancing understanding of complex geometric structures.

Contribution

It establishes the existence of strongly q-convex functions in neighborhoods of subvarieties in coverings, under specific topological and geometric conditions, extending previous convexity results.

Findings

Existence of strongly q-convex exhaustion functions near subvarieties

Conditions on covering spaces to ensure convexity properties

Application to complex geometric analysis

Abstract

The main result is that, for any projective compact analytic subset A of dimension q>0 in a reduced complex space X, there is a neighborhood U of A such that, for any covering space Z of X in which the lifting B of A has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a smooth exhaustion function on Z which is strongly q-convex on the lifting of U outside a uniform neighborhood of the q-dimensional compact irreducible components B.