Pluricomplex Green Functions on Stein Manifolds and Certain Linear Topological Invariants

Abstract

In this paper, we explore the existence of pluricomplex Green functions for Stein manifolds from a functional analysis point of view. For a Stein manifold $M$, we will denote by $O(M)$ the Fr\'echet space of analytic functions on $M$ equipped with the topology of uniform convergence on compact subsets. In the first section, we examine the relationship between the existence of pluricomplex Green functions and the diametral dimension of $O(M)$. This led us to consider negative plurisubharmonic functions on $M$ with a nontrivial relatively compact sublevel set (semi-proper). In section 2, we characterize Stein manifolds possessing a semi-proper negative plurisubharmonic function through a local version of the linear topological invariant $\widetilde{\Omega }$, of D.Vogt. In section 3 we look into pluri-Greenian complex manifolds introduced by E.Poletsky. We show that a complex manifold is locally uniformly pluri-Greenian if and only if it is pluri-Greenian and give a characterization of locally uniformly uniformly pluri-Greenian Stein manifolds in terms of notions introduced in section 2.