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October, 2012 Order of Meromorphic Maps and Rationality of the Image Space
J. Math. Soc. Japan 64(4): 1169-1180 (October, 2012). DOI: 10.2969/jmsj/06441169


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.


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Junjiro NOGUCHI. Jörg WINKELMANN. "Order of Meromorphic Maps and Rationality of the Image Space." J. Math. Soc. Japan 64 (4) 1169 - 1180, October, 2012.


Published: October, 2012
First available in Project Euclid: 29 October 2012

zbMATH: 1279.32016
MathSciNet: MR2998920
Digital Object Identifier: 10.2969/jmsj/06441169

Primary: 32H30
Secondary: 14M20

Keywords: meromorphic map , order , rationality

Rights: Copyright © 2012 Mathematical Society of Japan

Vol.64 • No. 4 • October, 2012
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