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 Vogt. In Section 3, we look into pluri-Greenian and locally uniformly pluri-Greenian complex manifolds introduced by 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.M S C 2 0 2 0 32A70, 32U35 (primary), 46E10 (secondary)