Meromorphic mappings of a complete connected Kähler manifold into a projective space sharing hyperplanes

Abstract

Let $M$ be a complete Kähler 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.

30 pages. In this version, we correct the typos of the last version, extend the notion of "functions of small integration" for all non-negative functions into $[0;+\infty ]$, which are continuous outside an analytic set of co-dimension two and attain $+\infty$ only in a analytic thin set. This paper is accepted for publication in Complex Variable and Elliptic Equations (2020)

Cited by in corpus (1)

• Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on Kähler manifolds