On fields of meromorphic functions on neighborhoods of rational curves

TL;DR

The paper analyzes the field of meromorphic functions on a smooth complex surface $F$ containing a positive self-intersection rational curve. It develops a deformation-theoretic framework, introducing good neighborhoods via Douady/Savelyev-style constructions, to control the local geometry around the curve. Using Castelnuovo’s rationality criterion and vanishing results for holomorphic forms on these neighborhoods, it proves that the meromorphic function field $\mathcal{M}(F)$ can only be isomorphic to $\mathbb{C}$, $\mathbb{C}(T)$, or $\mathbb{C}(T_1,T_2)$. Consequently, the field of meromorphic functions does not yield finer invariants for classifying neighborhoods of rational curves with positive self-intersection, and any richer classification would require extra structure. The work integrates deformation theory, finite generation of function fields, and rationality criteria to achieve a sharp dichotomy for $\mathcal{M}(F)$.

Abstract

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the field of meromorphic functions on $F$ is isomorphic to the field of rational functions in one or two variables over $\mathbb C$.

Figures (1)

• Figure 1:

Theorems & Definitions (27)

• Proposition 1.1
• Proposition 1.2
• Proposition 3.1
• Proposition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof : Sketch of proof
• Lemma 3.5
• proof