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

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 $U$.