Compact complex surfaces with no nonconstant meromorphic functions

TL;DR

This paper demonstrates that within the moduli spaces of certain compact complex surfaces—tori, Hopf surfaces, and K3 surfaces—there exists a dense subset of surfaces lacking nonconstant meromorphic functions, extending classical examples.

Contribution

It proves the density of surfaces with no nonconstant meromorphic functions in the moduli spaces of three key types of complex surfaces.

Findings

Dense sets of such surfaces exist in each moduli space.

Classical examples by Siegel and Kodaira are part of these dense subsets.

The result extends understanding of meromorphic function absence in complex geometry.

Abstract

In 1949 Siegel gave an example of a complex two-torus with no nonconstant meromorphic functions. In 1964 Kodaira showed that compact complex surfaces with no nonconstant meromorphic must be of the following three types: tori, Hopf type surfaces with first Betti number equal to one, and K3 surfaces. In this paper we show that surfaces of these three types have a dense set of surfaces in their natural moduli spaces with no nonconstant meromorphic functions.