Maximal subextensions of plurisubharmonic functions

TL;DR

This paper investigates conditions under which (quasi-)plurisubharmonic functions can be extended from subdomains to entire compact Kähler manifolds, establishing bounds on Monge-Ampère mass that guarantee such extensions and analyzing their properties.

Contribution

It provides new criteria based on Monge-Ampère mass bounds for the existence and properties of maximal subextensions of plurisubharmonic functions on Kähler manifolds.

Findings

Bound on Monge-Ampère mass ensures subextension existence.

Maximal subextensions can have well-defined Monge-Ampère measures.

Counterexample showing limitations of subextensions in projective space.

Abstract

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact K\"ahler manifold. We prove that a precise bound on the complex Monge-Amp\`ere mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Amp\`ere measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Amp\`ere measure and the right bound on its Monge-Amp\`ere mass on the unit ball in $\C^n$ for which the maximal subextension to the complex projective space $\mb P_n$ does not have a globally well defined complex Monge-Amp\`ere measure.