Nevanlinna theory on complete K\"ahler manifolds with non-negative Ricci curvature

TL;DR

This paper extends Nevanlinna theory to meromorphic mappings on complete K"ahler manifolds with non-negative Ricci curvature, establishing new defect relations, unicity theorems, and a five-value theorem extension.

Contribution

It develops an equidistribution theory and a second main theorem for such manifolds, leading to novel unicity results and a five-value theorem extension.

Findings

Established a sharp defect relation in Nevanlinna theory.

Proved a five-value theorem for meromorphic mappings on K"ahler manifolds.

Derived unicity theorems for dominant meromorphic mappings.

Abstract

The paper develops an equidistribution theory of meromorphic mappings from a complete K\"ahler manifold with non-negative Ricci curvature into a complex projective manifold intersecting normal crossing divisors. When the domain manifolds are of maximal volume growth, one obtains a second main theorem with a refined error term. As a result, we prove a sharp defect relation in Nevanlinna theory. Furthermore, our results are applied to the propagation problems of algebraic dependence. As major consequences, we set up several unicity theorems for dominant meromorphic mappings on complete K\"ahler manifolds. In particular, we prove a five-value theorem on complete K\"ahler manifolds, which gives an extension of Nevanlinna's five-value theorem for meromorphic functions on $\mathbb C.$