On minimal kernels and Levi currents on weakly complete complex manifolds

DOI10.1090/PROC/15946zbMATH Open1503.32006arXiv2102.05328OpenAlexW3128428510MaRDI QIDQ5086939

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Publication date: 8 July 2022

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Abstract: A complex manifold $X$ is emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $phi$. The minimal kernels $Sigm{a}_X^k,kin[0,infty]$ (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 $Sigm{a}_X^k$'s, and give sufficient conditions for points in $Sigm{a}_X^k$ 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 $Sigm{a}_X^{i}nfty$, 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.

Full work available at URL: https://arxiv.org/abs/2102.05328

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