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Nonabelian Hodge isomorphisms for stacks and cohomological Hall algebras

Lucien Hennecart

TL;DR

This work completes the nonabelian Hodge theory for stacks by proving that the Borel–Moore homologies of the Dolbeault, Betti, and de Rham moduli stacks admit compatible CoHA structures that coincide under NAHT. It introduces the Hodge–Deligne moduli space as a unifying interpolation via λ-connections, and it establishes CoHA structures both relative to and absolute on these stacks, with a PBW-type description and canonical BPS algebras. The paper further develops affinized BPS Lie algebras and demonstrates their isomorphism across the three NAHT sides, extends the framework to genus-one and nonzero degree cases, and analyzes χ-independence phenomena in CoHAs and their modules, including ℓ-adic settings and Galois conjugation. Collectively, this yields a CoHA-enhanced, stacky version of NAHT for curves, providing a robust bridge between geometric representation theory, mixed Hodge theory, and nonabelian Hodge correspondences with potential extensions to higher dimensions and categorification.

Abstract

In this paper, we complete the nonabelian Hodge theory (NAHT) triangle of isomorphisms for stacks between the Borel-Moore homologies of the Dolbeault, Betti, and de Rham moduli stacks. We first explain how to realise the category of connections on a smooth projective curve as a subcategory of a $2$-Calabi-Yau dg-category satisfying some appropriate geometric conditions. Then, we define a cohomological Hall algebra (CoHA) product on the Borel-Moore homology of the stack of connections on a smooth projective curve. This allows us to not only compare the Borel-Moore homologies of the stacks at the relative and absolute levels for the three sides of NAHT, but also to compare their CoHA structures: they all coincide. To compare the Dolbeault and de Rham sides, we define a relative CoHA for the Hodge-Deligne moduli space parametrising $λ$-connections. The Betti and de Rham sides are compared using the (derived) Riemann-Hilbert correspondence. The comparison of the Borel-Moore homologies of the Dolbeault and Betti moduli stacks was previously considered by the author with Davison and Schlegel Mejia (without taking the cohomological Hall algebra structures into account). This paper completes this study and provides a CoHA enhancement of the classical NAHT for curves.

Nonabelian Hodge isomorphisms for stacks and cohomological Hall algebras

TL;DR

This work completes the nonabelian Hodge theory for stacks by proving that the Borel–Moore homologies of the Dolbeault, Betti, and de Rham moduli stacks admit compatible CoHA structures that coincide under NAHT. It introduces the Hodge–Deligne moduli space as a unifying interpolation via λ-connections, and it establishes CoHA structures both relative to and absolute on these stacks, with a PBW-type description and canonical BPS algebras. The paper further develops affinized BPS Lie algebras and demonstrates their isomorphism across the three NAHT sides, extends the framework to genus-one and nonzero degree cases, and analyzes χ-independence phenomena in CoHAs and their modules, including ℓ-adic settings and Galois conjugation. Collectively, this yields a CoHA-enhanced, stacky version of NAHT for curves, providing a robust bridge between geometric representation theory, mixed Hodge theory, and nonabelian Hodge correspondences with potential extensions to higher dimensions and categorification.

Abstract

In this paper, we complete the nonabelian Hodge theory (NAHT) triangle of isomorphisms for stacks between the Borel-Moore homologies of the Dolbeault, Betti, and de Rham moduli stacks. We first explain how to realise the category of connections on a smooth projective curve as a subcategory of a -Calabi-Yau dg-category satisfying some appropriate geometric conditions. Then, we define a cohomological Hall algebra (CoHA) product on the Borel-Moore homology of the stack of connections on a smooth projective curve. This allows us to not only compare the Borel-Moore homologies of the stacks at the relative and absolute levels for the three sides of NAHT, but also to compare their CoHA structures: they all coincide. To compare the Dolbeault and de Rham sides, we define a relative CoHA for the Hodge-Deligne moduli space parametrising -connections. The Betti and de Rham sides are compared using the (derived) Riemann-Hilbert correspondence. The comparison of the Borel-Moore homologies of the Dolbeault and Betti moduli stacks was previously considered by the author with Davison and Schlegel Mejia (without taking the cohomological Hall algebra structures into account). This paper completes this study and provides a CoHA enhancement of the classical NAHT for curves.
Paper Structure (94 sections, 64 theorems, 130 equations)

This paper contains 94 sections, 64 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Nonabelian Hodge correspondence
  3. Main results
  4. CoHA structure for the de Rham stack
  5. Hodge--Deligne moduli stack
  6. Compatibility of the CoHA structures
  7. Affinized BPS Lie algebras
  8. Non-zero degree versions
  9. $\chi$-independence phenomena
  10. More questions
  11. Stacky nonabelian Hodge isomorphisms for higher dimensional varieties
  12. Stacky nonabelian Hodge isomorphisms in cohomology
  13. Categorification of stacky NAHT
  14. Cohomological Hall algebras in families
  15. Conventions and notations
  16. ...and 79 more sections

Key Result

Theorem 1.2

We have a relative cohomological Hall algebra structure on $\underline{\mathscr{A}}^{\mathrm{dR}}(C)\coloneqq\mathtt{JH}_*^{\mathrm{dR}}\mathbb{D}\underline{\mathbf{Q}}_{\mathfrak{M}^{\mathrm{dR}}(C)}^{\mathrm{vir}}$. It induces an absolute cohomological Hall algebra structure on $\mathop{\mathrm{H}

Theorems & Definitions (118)

  • Theorem 1.2: $\subseteq$ Theorem \ref{['theorem:CoHAdeRham']}
  • Theorem 1.3: =Theorem \ref{['theorem:CoHAdeRham']}
  • Theorem 1.4: =Theorem \ref{['theorem:CoHAHodgeDeligne']}
  • Theorem 1.5: =Theorem \ref{['theorem:relativBPSalg']}
  • Corollary 1.6: $\subseteq$ Proposition \ref{['proposition:trivializationCoHA']}
  • Corollary 1.7
  • Theorem 1.8: =Corollary \ref{['corollary:reldRdolcoincide']}+Theorem \ref{['theorem:reldRBetticoincide']}
  • Corollary 1.9
  • Remark 1.10
  • Theorem 1.11: =Theorem \ref{['theorem:actionChernclassdet']}
  • ...and 108 more