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Topology of $\mathbb{G}_m$-actions and applications to the moduli of Higgs bundles

Andres Fernandez Herrero, Siqing Zhang

TL;DR

The paper studies how $\mathbb{G}_m$-actions control the topology of moduli spaces and stacks in the Higgs-bundle setting, with a focus on positive characteristic. It develops general tools for cohomology under contracting $\mathbb{G}_m$-actions, then applies them to relate the Dolbeault and de Rham moduli $\mathbb{M}_{Dol,G}^d(C')$ and $\mathbb{M}_{dR,G}^{pd}(C)$ via an isomorphism of $\ell$-adic cohomology rings, under the low-height assumption on $G$. The authors also prove that the cohomology of the nilpotent cones matches across Dolbeault and de Rham theories, correct a gap in prior GL_n results, and extend the framework to the cohomology of the stack of $G$-Hitchin pairs as well as criteria for very stable $G$-bundles using $\mathbb{G}_m$-equivariant geometry. Collectively, the results refine the cohomological Nonabelian Hodge correspondence to a ring-level isomorphism and provide robust topological tools for moduli problems with $\mathbb{G}_m$-actions, with implications for both arithmetic and geometric contexts.

Abstract

We explain some results concerning the topology of varieties and stacks equipped with an action of the multiplicative group $\mathbb{G}_m$. We apply these techniques to the moduli of Higgs bundles. Our main application is to upgrade the cohomological Nonabelian Hodge Theorem in positive characteristic to an isomorphism of cohomology rings compatible with cup product.

Topology of $\mathbb{G}_m$-actions and applications to the moduli of Higgs bundles

TL;DR

The paper studies how -actions control the topology of moduli spaces and stacks in the Higgs-bundle setting, with a focus on positive characteristic. It develops general tools for cohomology under contracting -actions, then applies them to relate the Dolbeault and de Rham moduli and via an isomorphism of -adic cohomology rings, under the low-height assumption on . The authors also prove that the cohomology of the nilpotent cones matches across Dolbeault and de Rham theories, correct a gap in prior GL_n results, and extend the framework to the cohomology of the stack of -Hitchin pairs as well as criteria for very stable -bundles using -equivariant geometry. Collectively, the results refine the cohomological Nonabelian Hodge correspondence to a ring-level isomorphism and provide robust topological tools for moduli problems with -actions, with implications for both arithmetic and geometric contexts.

Abstract

We explain some results concerning the topology of varieties and stacks equipped with an action of the multiplicative group . We apply these techniques to the moduli of Higgs bundles. Our main application is to upgrade the cohomological Nonabelian Hodge Theorem in positive characteristic to an isomorphism of cohomology rings compatible with cup product.
Paper Structure (11 sections, 10 theorems, 13 equations)

This paper contains 11 sections, 10 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions on derived categories of sheaves
  3. Some general results
  4. Cohomology of equivariant sheaves
  5. Connectedness and fixed points
  6. Applications to Non Abelian Hodge theory in positive characteristic
  7. Non Abelian Hodge Theory in positive characteristic
  8. The cohomology of the nilpotent cones
  9. Further Applications
  10. The cohomology of the stack of $G$-Hitchin pairs
  11. Very stable $G$-bundles

Key Result

Proposition 2.2

In context: A^1 retraction, let $E \in D^b_c(\mathcal{X}, \overline{\mathbb Q}_{\ell})$ be a complex of sheaves satisfying $H^*E \cong p_2^*E$, where $p_2: \mathbb{A}^1 \times \mathcal{X} \to \mathcal{X}$ denotes the second projection. Then, the natural morphism of derived global sections $i^*: R\Ga

Theorems & Definitions (27)

  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5: herrero2023meromorphic
  • Proposition 2.6
  • proof
  • Lemma 3.4
  • ...and 17 more