Normal Forms for Ordinary Differential Operators. III

Junhu Guo^ab, A. B. Zheglov<sup>cd

a</sup> Sino–Russian Mathematics Center, Peking University, Beijing, China
^b School of Mathematical Sciences, Peking University, Beijing, China
^c Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991 Russia
^d National Research University Higher School of Economics (HSE University), Moscow, Russia

Abstract: In this paper, which is a follow-up of the previous ones (arXiv: 2406.14414 [math.AG], arXiv: 2511.05117 [math.AG]), we extend the explicit parametrization of torsion-free rank $1$ sheaves on projective irreducible curves with vanishing cohomology groups obtained earlier to an analogous parametrization of torsion-free sheaves of arbitrary rank with vanishing cohomology groups on projective irreducible curves. As an illustration of our theorem we calculate an explicit example, namely, the parametrization of rank $2$ sheaves on the Weierstrass cubic curve.

Keywords: commuting differential operators, quantum integrable systems, Krichever correspondence, moduli spaces of torsion-free sheaves, groups and algebras in quantum theory and relations with integrable systems.

Received: May 5, 2025
Revised: August 1, 2025
Accepted: August 4, 2025

DOI: 10.4213/tm4483

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 119–152

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