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

Abstract

Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m(R_0)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\infty$). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if $k$ differentibility nondegenerate meromorphic mappings $f^1,\dots ,f^k$ of $M$ into ${\mathbb{P}}^n(\mathbb{C})(n\ge 2)$ satisfying the condition $(C_{\rho })$ and sharing few hyperplanes in subgeneral position regardless of multiplicity then $f^1\wedge \cdots \wedge f^k\equiv 0$. For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of $M$ into ${\mathbb{P}}^n(\mathbb{C})$ sharing $q(q\sim 2N-n+3+O(\rho ))$ hyperplanes in $N-$subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of ${\mathbb{C}}^m$ and extend some previous results for the case of mappings on Kähler manifold.

In this version, the non-integrated part is removed. The name of the paper is changed. The paper consists of 19 pages. This paper has been accepted for publication in Analele Stiintifice ale Universitatii Ovidius Constanta (2022). arXiv admin note: substantial text overlap with arXiv:1909.01849

References in corpus (2)

• Non-integrated defect relation for meromorphic maps from a Kähler manifold intersecting hypersurfaces in subgeneral of ${\mathbb{P}}^n(\mathbb{C})$
• Meromorphic mappings of a complete connected Kähler manifold into a projective space sharing hyperplanes