Order of Meromorphic Maps and Rationality of the Image Space

2011/03/30 by Junjiro Noguchi, Noguchi, Junjiro, Jörg Winkelmann +1
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Primary 32H30 #Secondary 14M20 #math.AG #math.CV #msc:14M20 #msc:32H30

paper · pdf · doi:10.48550/arxiv.1103.5822

10 pages; article

arxiv created 2011/03/30 · arxiv updated 2011/03/31

Abstract

Let ι: \C² \hookrightarrow S be a compactification of the two dimensional complex space \C². By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\citeko71) that S is a rational surface. Here we deal with a more general meromorphic map f: \Cⁿ → X into a compact complex manifold X of dimension n, whose differential df has generically rank n. Let ρ_f denote the order of f. We will prove that if ρ_f<2, then every global symmetric holomorphic tensor must vanish; in particular, \it 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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