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

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 $\mathbb C.$