## On minimal kernels and Levi currents on weakly complete complex manifolds

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- by Fabrizio Bianchi and Samuele Mongodi PDF
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## Abstract:

A complex manifold $X$ is *weakly complete* if it admits a continuous plurisubharmonic exhaustion function $\phi$. The minimal kernels $\Sigma _X^k, k \in [0,\infty ]$ (the loci where 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.

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## Additional Information

**Fabrizio Bianchi**- Affiliation: CNRS, Univ. Lille, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France
- MR Author ID: 1144484
- ORCID: 0000-0002-6720-3211
- Email: fabrizio.bianchi@univ-lille.fr
**Samuele Mongodi**- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Roberto Cozzi 55, I–20125 Milano, Italy
- MR Author ID: 1023148
- ORCID: 0000-0002-7231-6165
- Email: samuele.mongodi@unimib.it
- Received by editor(s): February 19, 2021
- Received by editor(s) in revised form: October 17, 2021, and December 5, 2021
- Published electronically: April 29, 2022
- Additional Notes: This work was supported by the Reseach in Pairs 2019 program of the CIRM (Centro Internazionale de Ricerca Matematica), Trento and the FBK (Fondazione Bruno Kessler). This project also received funding from the I-SITE ULNE (ANR-16-IDEX-0004 ULNE), the LabEx CEMPI (ANR-11-LABX-0007-01) and from the CNRS program PEPS JCJC 2019.
- Communicated by: Filippo Bracci
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 3927-3939 - MSC (2020): Primary 32E05, 32Q28, 32T35, 32U10, 32U40
- DOI: https://doi.org/10.1090/proc/15946
- MathSciNet review: 4446241