Core sets in Kähler manifolds

Abstract

The primary objective of this paper is to study core sets in the setting of m-subharmonic functions on the class of (non-compact) Kähler manifolds. Core sets are investigated in different aspects by considering various classes of plurisubharmonic functions. One of the crucial concepts in studying the structure of this kind of sets is the pseudoconcavity. In a more general way, we will have the structure of core defined with respect to the m-subharmonic functions, which we call m-core in our setting, in terms of m-pseudoconcave sets. In the context of m-subharmonic functions, we define m-harmonic functions and show that, in Cn(n≥2) and more generally in any Kähler manifold of dimension at least 2, m-harmonic functions are pluriharmonic functions for m≥2.