Hypersurface Convexity and Extension of K\"ahler Forms

TL;DR

This paper generalizes a result relating the extension of Kähler forms to the structure of the complement of certain compact sets in complex manifolds, providing new criteria for when these complements are unions of positive divisors.

Contribution

It extends Nemirovski's result to broader classes of compact sets and characterizes the extension of Kähler forms via sublevel sets of strictly plurisubharmonic functions.

Findings

$dd^c\

X\setminus K$ is a union of positive divisors under certain conditions

Extension of $dd^c\

Abstract

The following generalization of a result of S. Nemirovski is proved: if $X$ is either a projective or a Stein manifold and $K\subset X$ is a compact sublevel set of a strictly plurisubharmonic function $\varphi$ defined in a neighborhood of $K$, then $X\setminus K$ is a union of positive divisors if and only if $dd^c\varphi$ extends to a Hodge form on $X$. For an arbitrary compact subset $K\subsetneq X$, this gives that $X\setminus K$ is a union of positive divisors if and only if $K$ admits a neighbourhood basis of sublevel sets of strictly plurisubharmonic functions with the $dd^c$-extension property.