On minimal kernels and Levi currents on weakly complete complex manifolds

A complex manifold X is weakly complete if it admits a continuous plurisubharmonic exhaustion function φ. The minimal kernels ΣkX, k ∈ [0, ∞] (the loci where all Ck 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 ΣkX's, and give sufficient conditions for points in ΣkX to be in the support of some Levi current. When X is a surface and φ 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 Σ∞X, 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.