Normal forms for ordinary differential operators, III
Junhu Guo, A. B. Zheglov
TL;DR
This work extends the explicit parametrisation of torsion-free rank $1$ sheaves on projective irreducible curves to torsion-free rank $r$ sheaves with vanishing $H^0$ and $H^1$, using partially normalised normal forms of commuting differential operators as coordinates. It develops the necessary algebraic–geometric framework (spectral data, Schur pairs, and their embeddings) and proves a main parametrisation theorem (Theorem T:parametrisation) that links equivalence classes of these normal forms to isomorphism classes of rank $r$ spectral sheaves. The paper further provides a detailed, explicit rank $2$ example on a Weierstrass cubic curve, deriving normal forms for $L_6$ with respect to $L_4$ under self-adjoint and non-self-adjoint cases, and classifying the resulting spectral sheaves (e.g., $\mathcal O(q_1)\oplus \mathcal O(q_2)$, $\mathcal O(q)\otimes \mathcal A$, $\mathcal B_q$, $\mathcal S\oplus \mathcal S$, etc.) across ν-values. It also clarifies how isomorphic sheaves correspond to conjugate matrix presentations via transfer matrices, thereby connecting the new parametrisation to known descriptions from the Fourier–Mukai framework in the rank-2 setting. Overall, the results offer an explicit, coordinate-based view of the moduli of higher-rank spectral sheaves and illustrate the compatibility with established spectral-data descriptions.
Abstract
In this paper, which is a follow-up of our first paper "Normal forms for ordinary differential operators, I", we extend the explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups obtained earlier to analogous parametrisation of torsion free sheaves of arbitrary rank with vanishing cohomology groups on projective irreducible curves. As an illustration of our theorem we calculate one explicit example of such parametrisation, namely for rank two sheaves on a Weierstrass cubic curve.