Meromorphic mappings from a Kähler manifold into a projective space sharing different families of hyperplanes

Abstract

Let $M$ be a complete Kähler Manifold, whose universal covering is biholomorphic to a ball in $\mathbb{C}^m$. We prove that if two linearly nondegenerate meromorphic mappings $f$ and $g$ from $M$ into $\mathbb{P}^n(\mathbb{C})$ share two different families of hyperplanes $\{H_j\}_{j=1}^q$ and $\{L_j\}_{j=1}^q$ without multiplicity then there is a linear projective transformation $\mathcal L$ of $\mathbb{P}^n(\mathbb{C})$ into itself such that $\mathcal L(g)\equiv f$ and $\mathcal L(L_j)=H_j$ $(1\le j\le q)$ for $q$ large enough.