Quasiplurisubharmonic Green functions

TL;DR

This paper investigates quasiplurisubharmonic Green functions on compact Kähler manifolds, focusing on their existence, properties, and singularity behavior, with full characterizations in specific cases like projective spaces.

Contribution

It provides a comprehensive study of Green functions with poles on Kähler manifolds, including existence criteria and detailed characterizations in concrete examples.

Findings

Existence criteria for quasiplurisubharmonic Green functions.

Characterization of singularities at the pole.

Explicit descriptions in projective spaces.

Abstract

Given a compact K\"ahler manifold $X$, a quasiplurisubharmonic function is called a Green function with pole at $p\in X$ if its Monge-Amp\`ere measure is supported at $p$. We study in this paper the existence and properties of such functions, in connection to their singularity at $p$. A full characterization is obtained in concrete cases, such as (multi)projective spaces.