Order of Meromorphic Maps and Rationality of the Image Space
Junjiro Noguchi, J\"org Winkelmann

TL;DR
This paper explores the relationship between the order of meromorphic maps from complex spaces to compact complex manifolds and the rationality of the target space, revealing the significance of the Kähler condition in this context.
Contribution
It generalizes Kodaira's result by linking the order of meromorphic maps to the rationality of the target space, highlighting the role of Kähler conditions in value distribution theory.
Findings
If the order of the map is less than 2, all global symmetric holomorphic tensors vanish.
For a 2-dimensional Kähler target, the space is rational.
Counter-example with a Hopf surface shows non-Kähler spaces do not necessarily have rationality.
Abstract
Let be a compactification of the two dimensional complex space . By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\cite{ko71}) that is a rational surface. Here we deal with a more general meromorphic map into a compact complex manifold of dimension , whose differential has generically rank . Let denote the order of . We will prove that if , then every global symmetric holomorphic tensor must vanish; in particular, {\it if and is k\"ahler, then 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…
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Taxonomy
TopicsMeromorphic and Entire Functions · Holomorphic and Operator Theory
