2021/12/25 by Aydın Aytuna, Aytuna, Aydın
Mathematics · #32 #46 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2112.13212
openalex publication_date 2021/12/25 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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 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 \widetildeΩ, 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.