*preprint*

**Inserted:** 9 jul 2022

**Last Updated:** 9 jul 2022

**Year:** 2021

**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.