Nevanlinna theory for meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact complex manifold

TL;DR

This paper develops a Nevanlinna theory for meromorphic maps from submanifolds of complex Euclidean space to compact complex manifolds, improving defect relations and establishing a unicity theorem.

Contribution

It introduces a Nevanlinna theory for such maps, refines defect relation definitions, and proves a unicity theorem for meromorphic maps from Stein manifolds.

Findings

Constructed a Nevanlinna theory for meromorphic maps from polydiscs.

Improved the non-integrated defect relation for these maps.

Proved a unicity theorem for meromorphic maps from Stein manifolds.

Abstract

The purpose of this article is threefold. The first is to construct a Nevanlina theory for meromorphic mappings from a polydisc to a compact complex manifold. In particular, we give a simple proof of Lemma on logarithmic derivative for nonzero meromorphic functions on $\mathbb{C}^l.$ The second is to improve the definition of the non-integrated defect relation of H. Fujimoto \cite{F2} and to show two theorems on the new non-integrated defect relation of meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact complex manifold. The third is to give a unicity theorem for meromorphic mappings from a Stein manifold to a compact complex manifold.