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Let ι : C2 ↪→ 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 ([2]) 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 ρf denote the order of f . We will prove that if ρf < 2, then every global symmetric holomorphic tensor must vanish; in particular, if dimX = 2 and X is kahler, then X is a rational surface. Without the kahler 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 kahler or non-kahler condition makes a difference in the value distribution theory. 2010 Mathematics Subject Classification. Primary 32H30; Secondary 14M20.

Order of Meromorphic Maps and Rationality of the Image Space