The linear flows in the space of Krichever-Lax matrices over an algebraic curve

TL;DR

This paper investigates the conditions under which flows on the space of Krichever-Lax matrices, related to the Hitchin system, are linear, using cohomological techniques inspired by Griffiths' work.

Contribution

It provides a necessary and sufficient cohomological condition for the linearity of flows on Krichever-Lax matrices in the Hitchin system.

Findings

Established a cohomological criterion for flow linearity.

Connected the linearity condition to the geometry of the moduli space.

Extended Griffiths' techniques to the context of Krichever-Lax matrices.

Abstract

In \cite{kri02}, I. M. Krichever invented the space of matrices parametrizing the cotangent bundle of moduli space of stable vector bundles over a compact Riemann surface, which is named as the Hitchin system after the investigation \cite{hit87}. We study a necessary and sufficient condition for the linearity of flows on the space of Krichever-Lax matrices in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work \cite{grif85} by P. A. Griffiths.