Order of Meromorphic Maps and Rationality of the Image Space

TL;DR

This paper explores the relationship between the order of meromorphic maps from complex spaces to compact complex manifolds and the rationality of the target space, revealing the significance of the Kähler condition in this context.

Contribution

It generalizes Kodaira's result by linking the order of meromorphic maps to the rationality of the target space, highlighting the role of Kähler conditions in value distribution theory.

Findings

If the order of the map is less than 2, all global symmetric holomorphic tensors vanish.

For a 2-dimensional Kähler target, the space is rational.

Counter-example with a Hopf surface shows non-Kähler spaces do not necessarily have rationality.

Abstract

Let $\iota : \C^2 \hookrightarrow S$ be a compactification of the two dimensional complex space $\C^2$. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\cite{ko71}) that $S$ is a rational surface. Here we deal with a more general meromorphic map $f: \C^n \to X$ into a compact complex manifold $X$ of dimension $n$, whose differential $df$ has generically rank $n$. Let $\rho_f$ denote the order of $f$. We will prove that if $\rho_f<2$, then every global symmetric holomorphic tensor must vanish; in particular, {\it if $\dim X=2$ and $X$ is k\"ahler, then $X$ is a rational surface. Without the k\"ahler condition there is no such conclusion, as we will show by a counter-example using a Hopf surface.} This may be the first instance that the k\"ahler or non-k\"ahler condition makes a difference in the value distribution theory.