Meromorphic mappings on K\"{a}hler manifolds weakly sharing hyperplanes in $\mathbb P^n(\mathbb C)$

TL;DR

This paper investigates the uniqueness of meromorphic mappings from Kähler manifolds to projective space that share hyperplanes under weaker conditions, providing new results and improvements in the field of complex geometry.

Contribution

It introduces a weaker sharing condition for hyperplanes and extends the uniqueness and algebraic dependence results for meromorphic mappings satisfying condition (C_ρ).

Findings

Established new uniqueness theorems under weaker hyperplane sharing conditions.

Improved existing results on algebraic dependence of meromorphic mappings.

Extended the scope of conditions under which mappings are uniquely determined.

Abstract

In this paper, we study the uniqueness problem for linearly nondegenerate meromorphic mappings from a K\"{a}hler manifold into $\mathbb P^n(\mathbb C)$ satisfying a condition $(C_\rho)$ and sharing hyperplanes in general position, where the condition that two meromorphic mappings $f,g$ have the same inverse image for some hyperplanes $H$ is replaced by a weaker one that $f^{-1}(H)\subset g^{-1}(H)$. Moreover, we also give some improvements on the uniqueness problem and algebraic dependence problem of meromorphic mappings which share hyperplanes and satisfy $(C_\rho)$ conditions for different non-negative numbers $\rho$.