Generalized complex Stein manifold

TL;DR

This paper introduces the concept of generalized complex Stein manifolds, extending classical complex analysis results, developing an $L^2$ theory, and establishing embedding theorems that characterize these manifolds.

Contribution

It defines GC Stein manifolds, extends Cartan's theorems, develops an $L^2$ theory with $L$-plurisubharmonic functions, and proves embedding theorems for these manifolds.

Findings

Extension of Cartan's Theorems A and B to GC geometry

Development of an $L^2$ theory for $L$-plurisubharmonic functions

Existence of proper GH embeddings into $ ext{R}^{2n-2k} imes ext{C}^{2k+1}$

Abstract

We introduce the notion of a generalized complex (GC) Stein manifold and provide complete characterizations in three fundamental aspects. First, we extend Cartan's Theorem A and B within the framework of GC geometry. Next, we define $L$-plurisubharmonic functions and develop an associated $L^2$ theory. This leads to a characterization of GC Stein manifolds using $L$-plurisubharmonic exhaustion functions. Finally, we establish the existence of a proper GH embedding from any GC Stein manifold into $\mathbb{R}^{2n-2k} \times \mathbb{C}^{2k+1}$, where $2n$ and $k$ denote the dimension and type of the GC Stein manifold, respectively. This provides a characterization of GC Stein manifolds via GH embeddings. Several examples of GC Stein manifolds are given.