Order of Meromorphic Maps and Rationality of the Image Space

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.