A refined form of the second main theorem on complete non-positively curved K\"ahler manifolds

How to devise a second main theorem with best error terms is a central problem in the study of Nevanlinna theory. However, it seems difficult to be done for a general non-positively curved K"ahler manifold. Based on the work of A. Atsuji in Nevanlinna theory, we present a refined form of the second main theorem of meromorphic mappings on a general complete K"ahler manifold with non-positive sectional curvature using a good estimate. This result improves the error terms in the second main theorem obtained by A. Atsuji in 2018.