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A complex manifold $X$ is \emph{weakly complete} if it admits a continuousplurisubharmonic exhaustion function $\phi$. The minimal kernels $\Sigma_X^k, k\in [0,\infty]$ (the loci where are all $\mathcal{C}^k$ plurisubharmonicexhaustion functions fail to be strictly plurisubharmonic),introduced bySlodkowski-Tomassini, and the Levi currents, introduced by Sibony, are bothconcepts aimed at measuring how far $X$ is from being Stein. We compare thesenotions, 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 supportof 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,weprove the existence of a Levi current precisely supported on $\Sigma_X^\infty$,and give a classification of Levi currents on $X$. In particular,unless $X$ isa modification of a Stein space, every point in $X$ is in the support of someLevi current.

On minimal kernels and Levi currents on weakly complete complex manifolds