On minimal kernels and Levi currents on weakly complete complex manifolds

TL;DR

This paper explores the relationship between minimal kernels and Levi currents on weakly complete complex manifolds, establishing support properties and classifications that deepen understanding of their geometric structure.

Contribution

It compares minimal kernels and Levi currents, proves support relations, and classifies Levi currents on surfaces with analytic exhaustion functions.

Findings

Levi currents are supported on all minimal kernels.

Conditions are provided for points in minimal kernels to support Levi currents.

On surfaces with analytic exhaustion, Levi currents are precisely supported on the infinite minimal kernel.

Abstract

A complex manifold $X$ is \emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $\phi$. The minimal kernels $\Sigma_X^k, k \in [0,\infty]$ (the loci where are all $\mathcal{C}^k$ plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic),introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far $X$ is from being Stein. We compare these notions, prove that all Levi currents are supported by all the $\Sigma_X^k$'s, and give sufficient conditions for points in $\Sigma_X^k$ to be in the support of some Levi current. When $X$ is a surface and $\phi$ can be chosen analytic, building on previous work by the second author, Slodkowski, and Tomassini,we prove the existence of a Levi current precisely supported on $\Sigma_X^\infty$, and give a classification of Levi currents on $X$. In particular,unless $X$ is a modification of a Stein space, every point in $X$ is in the support of some Levi current.