Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Normal forms for ordinary differential operators, I

J. Guo, A. B. Zheglov

TL;DR

This work addresses parametrising torsion-free rank-one sheaves with vanishing cohomology on projective curves via spectral data of rank-one commutative subrings of differential operators. It extends generalized Schur theory to dimension one, introducing a robust HCPC framework and normal forms for differential operators through Schur conjugation, with detailed control of centralizers. The central results describe the centralizer structure, embed normal forms into a skew pseudo-differential setting, and derive a concrete, affine-parameter description of the moduli space of spectral data as an open subset of the compactified Jacobian. The approach yields an explicit parametrisation of rank-one spectral sheaves and a template for higher-rank/higher-dimensional generalisations, supported by illustrative examples linking operator data to spectral curves.

Abstract

In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups. This parametrisation is obtained with the help of normal forms - a notion we introduce in this paper. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete non-commutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for any given commuting operators.

Normal forms for ordinary differential operators, I

TL;DR

This work addresses parametrising torsion-free rank-one sheaves with vanishing cohomology on projective curves via spectral data of rank-one commutative subrings of differential operators. It extends generalized Schur theory to dimension one, introducing a robust HCPC framework and normal forms for differential operators through Schur conjugation, with detailed control of centralizers. The central results describe the centralizer structure, embed normal forms into a skew pseudo-differential setting, and derive a concrete, affine-parameter description of the moduli space of spectral data as an open subset of the compactified Jacobian. The approach yields an explicit parametrisation of rank-one spectral sheaves and a template for higher-rank/higher-dimensional generalisations, supported by illustrative examples linking operator data to spectral curves.

Abstract

In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups. This parametrisation is obtained with the help of normal forms - a notion we introduce in this paper. Namely, considering the ring of ordinary differential operators as a subring of a certain complete non-commutative ring , the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for any given commuting operators.
Paper Structure (9 sections, 37 theorems, 179 equations)

This paper contains 9 sections, 37 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. List of notations
  3. Generic theory of normal forms
  4. Preliminary statements from the Schur theory for the ring $\hat{D}_1^{sym}$
  5. Basic formulae in $\hat{D}_1^{sym}$
  6. Homogeneous canonical polynomials
  7. Some necessary conditions on the Schur operator
  8. Some necessary conditions on normal forms
  9. Normal forms for commuting operators

Key Result

Theorem 1.1

Let $P_1, \ldots , P_n\in \hat{D}_n^{sym}$ be commuting operators with $\mathop {\rm \bf ord} (P_i)=k$ for all $i=1,\ldots ,n$. Assume that the module $F$ of the ring $K[\sigma (P_1), \ldots , \sigma (P_n)]$ is finitely generated and free. Then there exists an invertible operator $S\in \hat{D}_n^{sy

Theorems & Definitions (100)

  • Theorem 1.1: A generalized Schur theorem, Th. 7.1
  • Theorem 1.2: A centralizer theorem, Prop.7.1
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • ...and 90 more