Similarity of holomorphic matrices on 1-dimensional Stein spaces

TL;DR

This paper revises the proof of a theorem on the global holomorphic similarity of matrices on 1-dimensional Stein spaces, extending previous results from Riemann surfaces and suggesting applicability to higher dimensions.

Contribution

The paper provides a revised proof of the similarity theorem for holomorphic matrices on 1-dimensional Stein spaces, with potential extension to higher dimensions.

Findings

Revised proof of the similarity theorem for 1-dimensional Stein spaces.

Method applicable to higher-dimensional Stein spaces.

Extension of Guralnick's result to more general complex spaces.

Abstract

R. Guralnick [Linear Algebra Appl. 99, 85-96 (1988)] proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.