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$P=W$ via $\mathcal{H}_2$

Tamas Hausel, Anton Mellit, Alexandre Minets, Olivier Schiffmann

TL;DR

The paper resolves the $P=W$ conjecture for moduli spaces of stable Higgs bundles and stable parabolic Higgs bundles by constructing a natural action of the Lie algebra $\mathcal{H}_2$ on $H^*(X)[x,y]$ via tautological classes and Hecke correspondences. A key innovation is identifying an $\mathfrak{sl}_2$-triple acting on the tautological cohomology through a degeneration from a deformed to a rational Cherednik/CoHA framework, enabling a Lefschetz-style control of the perverse filtration. The authors show that the perverse filtration on $H^*(M)$ coincides with the $\mathfrak{sl}_2$-filtration induced by this triple, yielding $P=W$ for the elliptic locus and, through a sequence of restriction and nilpotent degeneration arguments, for the full Higgs moduli spaces including the parabolic case. The work leverages the cohomological Hall algebra of zero-dimensional sheaves, the Fock-space representation, and a refined reduction to a Weyl subalgebra to realize a robust $\mathcal{H}_2$-action, with implications for mirror symmetry and the broader structure of tautological cohomology on Higgs moduli.

Abstract

Let $\mathcal{H}_2$ be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let $X$ be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let $H^*(X)$ be the cohomology with rational coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of $\mathcal{H}_2$ on $H^*(X)[x,y]$, where $x$ and $y$ are formal variables. We show that the perverse filtration on $H^*(X)$ coincides with the filtration canonically associated to $\mathfrak{sl}_2\subset \mathcal{H}_2$ and deduce the $P=W$ conjecture of de Cataldo-Hausel-Migliorini.

$P=W$ via $\mathcal{H}_2$

TL;DR

The paper resolves the conjecture for moduli spaces of stable Higgs bundles and stable parabolic Higgs bundles by constructing a natural action of the Lie algebra on via tautological classes and Hecke correspondences. A key innovation is identifying an -triple acting on the tautological cohomology through a degeneration from a deformed to a rational Cherednik/CoHA framework, enabling a Lefschetz-style control of the perverse filtration. The authors show that the perverse filtration on coincides with the -filtration induced by this triple, yielding for the elliptic locus and, through a sequence of restriction and nilpotent degeneration arguments, for the full Higgs moduli spaces including the parabolic case. The work leverages the cohomological Hall algebra of zero-dimensional sheaves, the Fock-space representation, and a refined reduction to a Weyl subalgebra to realize a robust -action, with implications for mirror symmetry and the broader structure of tautological cohomology on Higgs moduli.

Abstract

Let be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let be the cohomology with rational coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of on , where and are formal variables. We show that the perverse filtration on coincides with the filtration canonically associated to and deduce the conjecture of de Cataldo-Hausel-Migliorini.
Paper Structure (43 sections, 49 theorems, 216 equations)

This paper contains 43 sections, 49 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. The $P=W$ conjecture.
  3. The algebra $\mathcal{H}_2$.
  4. The main construction
  5. Some remarks
  6. Topological prerequisites
  7. Correspondences
  8. Purity
  9. Diagonal class
  10. Virtual fundamental class
  11. Relative Lefschetz theory, perverse filtration and Hecke operators
  12. Relative Lefschetz theory
  13. Functoriality
  14. Hecke correspondences
  15. Notations
  16. ...and 28 more sections

Key Result

Theorem 1

The subspace $P_m H^*(M_{r,d})$ is the span of products of tautological classes of total weight $\leq 2(m + N)$, where $N=(g-1)r^2+1$.

Theorems & Definitions (89)

  • Conjecture : $P=W$, cataldo2012topology
  • Theorem : $P=C$ conjecture
  • Corollary
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 79 more