Degeneracy theorems for meromorphic mappings of a complete Kähler manifold sharing hyperplanes in a projective space

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 $q(q>C+\rho K)$ 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 ${\mathbb{P}}^n(\mathbb{C})$.

arXiv commentIn this version, we shorten the proofs. The paper consists of 14 pages. This paper has been accepted for publication in Publicationes Mathematicae Debrecen. arXiv admin note: text overlap with arXiv:1909.01849. text overlap with arXiv:2004.06705