### A Second Main Theorem on Parabolic Manifolds

Min Ru, Julie Tzu-Yueh Wang
2005 Asian Journal of Mathematics
In [St], [WS], Stoll and Wong-Stoll established the Second Main Theorem of meromorphic maps f : M → P N (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 P N (C), depending only on the given
more » ... only on the given hyperplanes, such that for every (possibly degenerate) meromorphic map f : M → P N (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. In [WS], W. Stoll and Pit-Mann Wong established the Second Main Theorem of meromorphic maps f : M → P N (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. Motivated by the works of Vojta (see [Vo2] , [Vo3]), we show that, for a finite set of hyperplanes in P N (C), there exists a finite union of proper linear subspaces, depending only on the given hyperplanes, such that for every meromorphic map f : M → P N (C), if its image is not contained in that union, then the inequality of Wong-Stoll's theorem still holds, except that the ramification term is lost. Here the exceptional subspaces(i.e. the subspaces which the image f (M ) is not contained in) depend only on the given hyperplanes and can be determined explicitly. We note that the Second Main Theorem for linearly degenerated maps was also studied by W.X. Chen(see [Chen]). The estimate in the Theorem of Chen holds without exceptions. However, his estimate is weaker than the estimate of the current paper(which allows a finite number exceptions). Throughout this paper, we shall use the standard notation in the value distribution theory of meromorphic maps on parabolic manifolds (see [WS] or [St]). An affine algebraic manifold M can be represented as a finite branch cover over C m , π : M → C m . Let κ be the sheet number of the projection π and d π be the degree of the branching divisor of τ . Our main theorem is stated as follows: Main Theorem. Let M be an affine algebraic manifold of complex dimension m. Let π : M → C m be a finite branched covering. Let H = {H 1 , ..., H q } be a finite collection of hyperplanes in P N (C) in general position. Then there exists a *