A Second Main Theorem on Parabolic Manifolds

Abstract

In [St], [WS], Stoll and Wong-Stoll established the Second Main Theorem of meromorphic maps $f: M \rightarrow {\Bbb P}^N({\Bbb C})$ intersecting hyperplanes, under the assumption that $f$ is linear non-degenerate, where $M$ is a $m$-dimensional affine algebraic manifold(the proof actually works for more general category of Stein parabolic manifolds). This paper deals with the degenerate case. Using P. Vojta's method, we show that there exists a finite union of proper linear subspaces of ${\Bbb P}^N({\Bbb C})$, depending only on the given hyperplanes, such that for every (possibly degenerate) meromorphic map $f: M \rightarrow {\Bbb P}^N({\Bbb C})$, if its image is not contained in that union, the inequality of Wong-Stoll's theorem still holds (without the ramification term). We also carefully examine the error terms appearing in the inequality.