Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants

Tasuki Kinjo, Naoki Koseki

TL;DR

The paper proves χ-independence for Gopakumar–Vafa invariants of local curves using a global d-critical chart description that recasts moduli spaces as critical loci, and introduces BPS cohomology for Dolbeault moduli with noncoprime χ, showing χ-independence via mixed Hodge module isomorphisms on the Hitchin base. It develops a cohomological integrality framework for twisted Higgs bundles, including an approximation-by-proper-morphisms method and monodromic mixed Hodge modules, yielding a symmetric-product decomposition of vanishing-cycle data. The Higgs-bundle case is then analyzed through dimensional reduction, establishing χ-independence and integrality for BPS cohomology, with explicit examples illustrating differences between intersection cohomology and BPS cohomology in noncoprime settings. Together, these results extend P=W-type phenomena to broader noncoprime contexts, provide a unifying cohomological perspective on GV/BPS invariants, and deepen the link between curve counting and Higgs-bundle moduli via d-critical geometry and mixed Hodge theory.

Abstract

The aim of this paper is two-fold: Firstly, we prove Toda's $χ$-independence conjecture for Gopakumar--Vafa invariants of arbitrary local curves. Secondly, following Davison's work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank $r$ and Euler characteristic $χ$ which are not necessary coprime, and show that it does not depend on $χ$. This result extends the Hausel--Thaddeus conjecture on the $χ$-independence of E-polynomials proved by Mellit, Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.

Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants

TL;DR

The paper proves χ-independence for Gopakumar–Vafa invariants of local curves using a global d-critical chart description that recasts moduli spaces as critical loci, and introduces BPS cohomology for Dolbeault moduli with noncoprime χ, showing χ-independence via mixed Hodge module isomorphisms on the Hitchin base. It develops a cohomological integrality framework for twisted Higgs bundles, including an approximation-by-proper-morphisms method and monodromic mixed Hodge modules, yielding a symmetric-product decomposition of vanishing-cycle data. The Higgs-bundle case is then analyzed through dimensional reduction, establishing χ-independence and integrality for BPS cohomology, with explicit examples illustrating differences between intersection cohomology and BPS cohomology in noncoprime settings. Together, these results extend P=W-type phenomena to broader noncoprime contexts, provide a unifying cohomological perspective on GV/BPS invariants, and deepen the link between curve counting and Higgs-bundle moduli via d-critical geometry and mixed Hodge theory.

Abstract

The aim of this paper is two-fold: Firstly, we prove Toda's -independence conjecture for Gopakumar--Vafa invariants of arbitrary local curves. Secondly, following Davison's work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank and Euler characteristic which are not necessary coprime, and show that it does not depend on . This result extends the Hausel--Thaddeus conjecture on the -independence of E-polynomials proved by Mellit, Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.
Paper Structure (34 sections, 46 theorems, 227 equations)

This paper contains 34 sections, 46 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Motivation and Results
  3. Non-abelian Hodge theory
  4. Gopakumar--Vafa (BPS) invariants
  5. Strategy of the proof
  6. Results on local curves
  7. Results on Higgs bundles
  8. Relation with existing works
  9. Structure of the paper
  10. Preliminaries
  11. D-critical structures
  12. Vanishing cycle complexes on d-critical stacks
  13. Maulik--Toda's construction of Gopakumar--Vafa invariants
  14. Local curves and twisted Higgs bundles
  15. Spectral correspondence for local curves
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $r, m, m'$ be integers such that $r>0$ and $\gcd(r, m) = \gcd(r, m') = 1$ hold. Then there exists an isomorphism preserving the Hodge structure and the perverse filtration.

Theorems & Definitions (106)

  • Theorem 1.1: Example \ref{['ex:ICvsBPS']}
  • Theorem 1.2: Corollary \ref{['cor:chiBPS']}
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5: Corollary \ref{['cor:chiBM']}
  • Conjecture 1.6: Tod17
  • Theorem 1.7: Theorem \ref{['thm:chi-lC']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • ...and 96 more