Pluricomplex Green Functions on Stein Manifolds and Certain Linear Topological Invariants

TL;DR

This paper investigates the existence and properties of pluricomplex Green functions on Stein manifolds, linking them to linear topological invariants and providing characterizations of various pluri-Greenian conditions.

Contribution

It establishes new connections between pluricomplex Green functions, the diametral dimension of analytic function spaces, and linear topological invariants on Stein manifolds.

Findings

Existence of pluricomplex Green functions relates to diametral dimension of $O(M)$

Characterization of Stein manifolds with semi-proper negative plurisubharmonic functions

Equivalence of local uniform pluri-Greenian and pluri-Greenian properties

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.