Uniqueness problem with truncated multiplicities in value distribution theory. {II}

Abstract

Let $H_{1}, H_{2}, \ldots, H_{q}$ be hyperplanes in $P^{N}(\mathbb{C})$ in general position. Previously, the author proved that, in the case where $q \geq 2N+3$, the condition $\nu(f, H_{j}) = \nu(g, H_{j})$ imply $f = g$ for algebraically nondegenerate meromorphic maps $f, g : \mathbb{C}^{n} \to P^{N}(\mathbb{C})$, where $\nu(f, H_{j})$ denote the pull-backs of $H_{j}$ through $f$ considered as divisors. In this connection, it is shown that, for $q \geq 2N+2$, there is some integer $\ell_{0}$ such that, for any two nondegenerate meromorphic maps $f, g : \mathbb{C}^{n} \to P^{N}(\mathbb {C})$ with $\min(\nu(f, H_{j}), \ell_{0}) = \min(\nu(g, H_{j}), \ell_{0})$ the map $f \times g$ into $P^{N}(\mathbb{C}) \times P^{N}(\mathbb{C})$ is algebraically degenerate. He also shows that, for $N = 2$ and $q = 7$, there is some $\ell_{0}$ such that the conditions $\min(\nu(f, H_{j}), \ell_{0}) = \min(\nu(g, H_{j}), \ell_{0})$ imply $f = g$ for any two nondegenerate meromorphic maps $f, g$ into $P^{2}(\mathbb{C})$ and seven generic hyperplanes $H_{j}$'s.