Minimal kernels of weakly complete spaces

Abstract

Let X be a weakly complete space i.e. X a complex space endowed with a C k -smooth, k⩾0, plurisubharmonic exhaustion function. We give the notion of minimal kernel Σ 1= Σ 1( X) of X by the following property: x∈ Σ 1 if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ 1 of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ϕ, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ϕ a minimal function for X, the nonempty level sets Σ c 1= Σ 1∩{ ϕ= c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim c X=2 and c is a regular value of a minimal function ϕ then the nonempty level sets Σ c 1= Σ 1∩{ ϕ= c} are compact spaces foliated by holomorphic curves.