Erratum: Some Remarks on the Cohomology of Krichever-Novikov Algebras

TL;DR

This paper corrects a previous result on the cohomology of Krichever-Novikov algebras, clarifies the true lemma, and discusses an open problem about approximating holomorphic vector fields on Riemann surfaces.

Contribution

It identifies an error in the main lemma of prior work and states the correct version, highlighting an open problem in approximation theory on Riemann surfaces.

Findings

The main lemma in the original work is only partially true.

The correct version of the lemma is provided.

An open problem about approximation of vector fields remains unsolved.

Abstract

We discovered that only a weakened version of the main lemma is true. We state the right version, and the remaining open problem: Is it possible to approximate holomorphic vector fields (or more generally, sections in a line bundle) on an open Riemann surface of finite type (i.e. a compact one without a finite number of points) by meromorphic vector fields (where the poles are supposed to be in the distinguished points) ? We know that this is true for functions.