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The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations

Tomás L. Gómez, Andres Fernandez Herrero, Alfonso Zamora

Abstract

Let X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation $ρ$ of G into a product of general linear groups, we define a moduli stack of principal $ρ$-sheaves that compactifies the stack of G-bundles on X. We apply the theory developed by Alper, Halpern-Leistner and Heinloth to construct a moduli space of Gieseker semistable principal $ρ$-sheaves. This provides an intrinsic stack-theoretic construction of the moduli space of semistable singular principal bundles as constructed by Schmitt and Gómez-Langer-Schmitt-Sols. Our second main result is the definition of a schematic Gieseker-Harder-Narasimhan filtration for $ρ$-sheaves, which induces a stratification of the stack by locally closed substacks. This filtration for a general reductive group G is a refinement of the canonical slope parabolic reductions previously considered at the level of points by Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure. In an appendix, we apply the same techniques to define Gieseker-Harder-Narasimhan filtrations in arbitrary characteristic and show that they induce a stratification of the stack by radicial morphisms.

The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations

Abstract

Let X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation of G into a product of general linear groups, we define a moduli stack of principal -sheaves that compactifies the stack of G-bundles on X. We apply the theory developed by Alper, Halpern-Leistner and Heinloth to construct a moduli space of Gieseker semistable principal -sheaves. This provides an intrinsic stack-theoretic construction of the moduli space of semistable singular principal bundles as constructed by Schmitt and Gómez-Langer-Schmitt-Sols. Our second main result is the definition of a schematic Gieseker-Harder-Narasimhan filtration for -sheaves, which induces a stratification of the stack by locally closed substacks. This filtration for a general reductive group G is a refinement of the canonical slope parabolic reductions previously considered at the level of points by Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure. In an appendix, we apply the same techniques to define Gieseker-Harder-Narasimhan filtrations in arbitrary characteristic and show that they induce a stratification of the stack by radicial morphisms.

Paper Structure

This paper contains 30 sections, 53 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Stack-theoretic approach vs. GIT
  3. Preliminaries
  4. Notation
  5. Big open sets and torsion-free sheaves
  6. The scheme of $G$-reductions
  7. Principal $\rho$-sheaves
  8. Principal $\rho$-sheaves and the moduli stack $\text{Bun}_{\rho}(X)$
  9. Filtrations of $\rho$-sheaves
  10. Filtrations of $\rho$-sheaves in the split case
  11. $\Theta$-stratification on $\mathop{\mathrm{Bun}}\nolimits_{\rho}(X)$
  12. The numerical invariant
  13. Affine Grassmannians for $\rho$-sheaves
  14. Monotonicity of the numerical invariant
  15. HN boundedness
  16. ...and 15 more sections

Key Result

Theorem 1

Fix a central faithful representation $\rho: G \hookrightarrow \prod_{i} \mathop{\mathrm{GL}}\nolimits_{n_i}$. Then

Theorems & Definitions (161)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 151 more