Degeneracy theorems for meromorphic mappings of a complete K\"{a}hler manifold sharing hyperplanes in a projective space

Abstract

Let $M$ be a complete K\"{a}hler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_\rho)$ and sharing $q\ (q> C+\rho K)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C <2n$ (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for $k\ (2\le k\le n+1)$ meromorphic mappings sharing hyperplanes is also given. Our result generalize the previous result in the case where the mappings from $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$.