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.

Bianchi, F., Mongodi, S. (2022). On minimal kernels and Levi currents on weakly complete complex manifolds. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 150(9), 3927-3939 [10.1090/proc/15946].

On minimal kernels and Levi currents on weakly complete complex manifolds

Mongodi S.
2022

Abstract

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.
Articolo in rivista - Articolo scientifico
minimal kernels; levi currents; psh exhaustion
English
2022
150
9
3927
3939
none
Bianchi, F., Mongodi, S. (2022). On minimal kernels and Levi currents on weakly complete complex manifolds. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 150(9), 3927-3939 [10.1090/proc/15946].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/397631
Citazioni
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 4
Social impact