m-pseudoconcavity and compactness

The core of a compact set in a general complex manifold has beendefined by Shcherbina very recently to study the existence of strictly plurisubharmonic functions on compact sets. In this paper, using m-subharmonic functionson compact subsets of a non-compact Kähler manifold, we define the set m-coreof a compact set and investigate the structure of it.

We will have the decomposition of the m-minimal kernel of a weaklym-complete manifold and show that it can be fully decomposed into compactm-pseudoconcave subsets via certain results obtained in the author’s very recentpapers to have the disintegration of the set m-core of the entire Kähler manifold(or of a domain in the manifold) and to study the characterization of so-calledm-Stein manifolds.