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Hitchin maps and parabolic Hitchin maps on the moduli spaces of Hitchin sheaves on nodal curves

Usha N. Bhosle

TL;DR

This work extends Hitchin map theory to $L$-twisted Hitchin and parabolic Hitchin sheaves on a nodal integral curve $Y$, developing the moduli of Hitchin pairs and their parabolic variants, and establishing the Hitchin base, fibres, and BNR-type correspondences via spectral curves $Y_a$, $Y^p_a$, and $X^p_a$. It proves the properness of Hitchin morphisms, characterises fibres for general and special bases, and provides detailed analyses of nilpotent cones and very stable (parabolic) bundles, including the fixed-determinant case. A key contribution is the parabolic BNR correspondence and its strongly parabolic refinement, together with the construction and semistability of the parabolic Poincaré sheaf in coprime situations, all in the nodal setting. The results generalise known smooth-curve stories to singular curves, offering new tools for understanding Higgs-type moduli in singular contexts and laying groundwork for potential Langlands-type applications in singular or parabolic geometries.

Abstract

We study the Hitchin maps on the moduli spaces of Hitchin sheaves and parabolic Hitchin sheaves on a nodal integral curve $Y$. We study their fibres, the BNR correspondences and the relation of the restriction of the Hitchin map with very stable bundles. As an application, in the "coprime" case we prove that the parabolic Poincaré sheaf on $U_{par}(n, ξ)\times Y$ is parabolic stable where $U_{par}(n, ξ)$ is a fixed determinant moduli space of parabolic sheaves on $Y$.

Hitchin maps and parabolic Hitchin maps on the moduli spaces of Hitchin sheaves on nodal curves

TL;DR

This work extends Hitchin map theory to -twisted Hitchin and parabolic Hitchin sheaves on a nodal integral curve , developing the moduli of Hitchin pairs and their parabolic variants, and establishing the Hitchin base, fibres, and BNR-type correspondences via spectral curves , , and . It proves the properness of Hitchin morphisms, characterises fibres for general and special bases, and provides detailed analyses of nilpotent cones and very stable (parabolic) bundles, including the fixed-determinant case. A key contribution is the parabolic BNR correspondence and its strongly parabolic refinement, together with the construction and semistability of the parabolic Poincaré sheaf in coprime situations, all in the nodal setting. The results generalise known smooth-curve stories to singular curves, offering new tools for understanding Higgs-type moduli in singular contexts and laying groundwork for potential Langlands-type applications in singular or parabolic geometries.

Abstract

We study the Hitchin maps on the moduli spaces of Hitchin sheaves and parabolic Hitchin sheaves on a nodal integral curve . We study their fibres, the BNR correspondences and the relation of the restriction of the Hitchin map with very stable bundles. As an application, in the "coprime" case we prove that the parabolic Poincaré sheaf on is parabolic stable where is a fixed determinant moduli space of parabolic sheaves on .
Paper Structure (44 sections, 52 theorems, 237 equations)

This paper contains 44 sections, 52 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. The nilpotent cone for $Y$
  3. The algebraic stack $Vect_{Y,n}$
  4. The algebraic stacks $Vect^{\lambda_{\bullet}}_{Y, \nu_{\bullet}}$ and $Tf^{\lambda_{\bullet}}_{Y, \nu_{\bullet}}$
  5. The moduli spaces and the Hitchin map
  6. The Hitchin map
  7. The set map $h_{set}$
  8. Generalised Parabolic Hitchin pairs (GPH) on an integral curve $Y$
  9. Good GPH
  10. The Hitchin morphism $h: N_Y(n,d,L) \to {\bf A}$
  11. The fibre of the Hitchin map over a general $a \in {\bf A}'$
  12. Spectral curve and Hitchin Fibre over a general $a \in {\bf A}$
  13. Examples
  14. The closed subsets ${\bf H}(n,d,\omega_Y)$ and ${\bf H}(n,d,L), d(L) \ge 2g-1$.
  15. Very stable torsion-free sheaves
  16. ...and 29 more sections

Key Result

Theorem 1.1

For every nilpotent type $(\nu_{\bullet}, \lambda_{\bullet})$, the stack $\Lambda'_{Y, (\nu_{\bullet}, \lambda_{\bullet})} \subset T^* Vect_{Y,n}$ is either empty or smooth over $\mathbb{C}$, of pure dimension $n^2(g-1)$ and Lagrangian in $T^* Vect_{Y,n}$.

Theorems & Definitions (112)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 102 more