Meromorphic mappings of a complete connected K\"{a}hler manifold into a projective space sharing hyperplanes

TL;DR

This paper proves new uniqueness theorems for meromorphic mappings from certain complete Kähler manifolds into projective space, showing that sharing enough hyperplanes in general position implies the mappings are identical.

Contribution

It extends classical results on meromorphic mappings from complex Euclidean spaces to more general Kähler manifolds, establishing conditions for uniqueness based on hyperplane sharing.

Findings

If three meromorphic mappings share enough hyperplanes, then two of them are equal.

Sharing hyperplanes with multiplicity counted to level n+1 implies all three mappings are identical.

Results generalize known theorems from complex Euclidean spaces to Kähler manifolds.

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 $q\ (q> 2n+1+\alpha+\rho K)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $\alpha <1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to level $n+1$ then $f^1=f^2=f^3.$ Our results generalize the finiteness and uniqueness theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+2$ hyperplanes in general position with truncated multiplicity.