Uniqueness Theorems for Meromorphic Mappings with Few Targets

TL;DR

This paper establishes new uniqueness theorems for meromorphic mappings from complex Euclidean spaces to projective space, reducing the number of hyperplanes needed for uniqueness and generalizing recent results.

Contribution

It introduces explicit bounds on the number of hyperplanes for uniqueness, depending on a parameter c, and improves the coefficient in the hyperplane count formula.

Findings

Uniqueness theorems hold for q ewer hyperplanes than previously known.

The bounds depend explicitly on a parameter c and a threshold N(c).

The coefficient in the hyperplane count formula can be made smaller than 3 for large n.

Abstract

The purpose of this article is to show uniqueness theorems for meromorphic mappings of C^m to CP^n with few hyperplanes H_j, j=1,...,q. It is well known that uniqueness theorems hold for q \geq 3n+2. In this paper we show that for every nonnegative integer c there exists a positive integer N(c), depending only on c in an explicit way, such that uniqueness theorems hold if q\geq (3n+2 -c) and n\geq N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n>>0. At the same time, a big number of recent uniqueness theorems are generalized considerably.