Meromorphic maps from ${\Bbb C}^{p}$ into semi-abelian varieties and general projective varieties

TL;DR

This paper extends classical Nevanlinna theory results to meromorphic maps from complex Euclidean spaces into projective and semi-abelian varieties, using fiber integration to generalize key theorems.

Contribution

It introduces a method based on fiber integration to generalize Nevanlinna-type theorems for higher-dimensional complex spaces into projective varieties.

Findings

Extended Bloch's theorem to meromorphic maps from ${C}^p$

Generalized Second Main Theorem for semi-abelian varieties

Proved Second Main Theorem for maps into projective space with high-degree hypersurfaces

Abstract

In 1953, W. Stoll proposed a method of studying holomorphic functions of several complex variables by reducing them to one variable through fiber integration. In this paper, we use this method to extend some important Nevanlinna-type results for holomorphic curves into projective varieties to meromorphic maps from ${\Bbb C}^{p}$ to projective varieties. This includes Bloch's theorem and Noguchi-Winklemann-Yamanoi's Second Main Theorem for holomorphic maps into semi-abelian varieties intersecting an effective divisor, as well as Huynh-Vu-Xie's Second Main Theorem for meromorphic maps into projective space intersecting with a generic hypersurface with sufficiently high degree.