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 𝔹m(R0) in ℂm (0 < R0 +∞). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if k differentibility nonde-generate meromorphic mappings f1, …, fk of M into ℙn(ℂ) (n ≥ 2) satisfying the condition (Cρ) and sharing few hyperplanes in subgeneral position regardless of multiplicity then f1 Λ … Λ fk0. For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of M into ℙn(ℂ) sharing q (q ∼ 2N − n + 3 + O(ρ)) hyperplanes in N−subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of ℂm into ℙn(ℂ) and extend some previous results for the case of mappings on Kähler manifold.

Keywords

• algebraic dependence
• finiteness theorem
• kähler manifold
• meromorphic mapping
• primary 32h30, 32a22
• secondary 30d35