Stabilization of fields of meromorphic functions on neighborhoods of a rational curve

TL;DR

This paper proves that on certain complex surfaces containing a rational curve with positive self-intersection, meromorphic functions near the curve can be extended to a neighborhood, demonstrating a stabilization property.

Contribution

It establishes a stabilization result for meromorphic functions on neighborhoods of rational curves with positive self-intersection on complex surfaces.

Findings

Existence of a neighborhood where meromorphic functions extend globally

Extension property holds for neighborhoods of rational curves with positive self-intersection

Provides a new understanding of function extension in complex surface neighborhoods

Abstract

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve $C$ with positive self-intersection. We prove that there exists a neighborhood $U\supset C$ such that any meromorphic function defined on a connected neighborhood of $C$ in $U$ can be extended to a meromorphic function on the entire $U$.