Meromorphic Convexity on Complex Manifolds

TL;DR

This paper introduces the concept of meromorphic convexity on complex manifolds, defining a new class called M-manifolds that generalizes Stein and projective manifolds, with examples of nonconstant holomorphic functions.

Contribution

It defines meromorphic convexity on complex manifolds and introduces M-manifolds, expanding the understanding of global meromorphic functions on complex structures.

Findings

M-manifolds include all Stein and projective manifolds

Existence of noncompact M-manifolds without nonconstant holomorphic functions

Meromorphic convexity generalizes classical convexity notions

Abstract

The notion of meromorphic convexity is defined and studied on complex manifolds. Using this notion, in analogy with Stein manifolds, a new class of complex manifolds, called {\calligra M }-manifolds, is introduced. This is a class of complex manifolds with a good supply of global meromorphic functions, in particular, it includes all Stein manifolds and projective manifolds. It is also shown that there exist noncompact complex manifolds, known as long $\mathbb C^2$, that are {\calligra M }-manifolds but do not contain any nonconstant holomorphic functions.