Order of Meromorphic Maps and Rationality of the Image Space

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 ([2]) 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, if $\dim X=2$ and $X$ is kähler, then $X$ is a rational surface. Without the kähler 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 kähler or non-kähler condition makes a difference in the value distribution theory.