Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

TL;DR

This paper proves that the intersection cohomology of certain moduli spaces of sheaves and Higgs bundles is independent of the Euler characteristic, confirming conjectures in algebraic geometry and string theory contexts.

Contribution

It establishes cohomological $ ext{chi}$-independence for moduli of sheaves and Higgs bundles, extending previous conjectures and applying advanced geometric techniques.

Findings

Cohomology is independent of Euler characteristic for specified moduli spaces.

Confirms Bousseau's cohomological $ ext{chi}$-independence conjecture for $ ext{P}^2$.

Verifies Toda's conjecture for Gopakumar-Vafa invariants in certain cases.

Abstract

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $\chi$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces. For the proof, we combine a generalized version of Ng\^o's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.