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 Fréchet 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 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, controlled approximation type condition, which can be considered as a local version of the linear topological invariant Ω of D.Vogt. In Section 3 we look into pluri-Greenian and locally uniformly 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 pluri-Greenian Stein manifolds in terms of the notions introduced in Section 2.