A uniqueness theorem for meromorphic maps into $\mathbb{P}^n$ with generic $(2n+2)$ hyperplanes

TL;DR

This paper proves a uniqueness theorem for meromorphic maps from complex Euclidean spaces into projective space, showing that if two such maps share the same inverse images of a set of generic hyperplanes and one is algebraically non-degenerate, then they are identical.

Contribution

It provides a complete proof of a known uniqueness theorem for meromorphic maps into projective space with generic hyperplanes, clarifying previous implicit results.

Findings

Uniqueness of meromorphic maps under shared hyperplane inverse images

Extension of Fujimoto's results with a complete proof

Conditions ensuring maps are identical if sharing hyperplanes

Abstract

Let $ H_1,\dots,H_{2n+2}$ be \emph{generic} $(2n+2)$ hyperplanes in $\mathbb{P}^n.$ It is proved that if meromorphic maps $ f $ and $ g $ of $\mathbb{C}^m $ into $\mathbb{P}^n $ satisfy $ f^*(H_j)=g^*(H_j)$ $(1\leq j\leq 2n+2)$ and $ g $ is algebraically non-degenerate then $ f=g.$ This result is essentially implied by the proof of Hirotaka Fujimoto in papers [Nagoya Math. J., 1976(64): 117--147] and [Nagoya Math. J., 1978(71): 13--24]. This note gives a complete proof of the above uniqueness result.