Order of Meromorphic Maps and Rationality of the Image Space

Details Order of Meromorphic Maps and Rationality of the Image Space Order of Meromorphic Maps and Rationality of the Image Space

2011-03-30

arxiv.org/abs/1103.5822v1

Mathematics

Complex Variables

10 pages; article

Scientific paper

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.

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