A note on Fujimoto's uniqueness theorem with $(2n+3)$ hyperplanes

TL;DR

This paper revisits Fujimoto's uniqueness theorem for meromorphic maps sharing hyperplanes, identifies a mistake in the original proof, and offers a corrected, complete proof from a new perspective.

Contribution

It corrects a mistake in Fujimoto's original proof and provides a complete, clearer demonstration of the theorem using a new approach.

Findings

The correction clarifies the proof of Fujimoto's theorem.

The corrected proof confirms the theorem's validity.

The approach simplifies understanding of the uniqueness condition.

Abstract

Hirotaka Fujimoto proved in [Nagoya Math. J., 1976(64): 117--147] and [Nagoya Math. J., 1978(71): 13--24] a uniqueness theorem for algebraically non-degenerate meromorphic maps into $\mathbb{P}^n(\mathbb{C})$ sharing $(2n+3)$ hyperplanes in general position. The proof of Lemma 3.6 in [Nagoya Math. J., 1976(64): 117--147] is found to contain a mistake. This mistake can be corrected easily from a new point of view. This note explains the idea of the correction and provides a complete proof.