Order of Meromorphic Maps and Rationality of the Image Space

Abstract

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 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, 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.