Meromorphic Convexity on Complex Manifolds

TL;DR

This paper develops the theory of meromorphic convexity on complex manifolds and introduces $M$-manifolds as a meromorphic analogue of Stein manifolds. It defines meromorphic hulls $\widehat{K}_X$ and inner hulls $\widetilde{K}_X$, and establishes divisor-theoretic frameworks for meromorphic functions, including indeterminacy sets, to study convexity in the meromorphic setting. The authors prove Oka--Weil--type approximation theorems on non-Stein, meromorphically convex manifolds and construct examples showing long $\mathbb{C}^2$ that are $M$-manifolds, even when they admit no nonconstant holomorphic functions. They also provide a meromorphic Runge criterion giving sufficient conditions for long $\mathbb{C}^2$ to be $M$-manifolds, and present spreadability and representation results for meromorphic functions on $1$-convex spaces, linking to Moishezon theory and the projective setting.

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.

Theorems & Definitions (36)

• Definition 1
• Definition 2
• Definition 3
• Proposition 4
• proof
• Proposition 5
• proof
• Definition 6
• Proposition 7
• proof