Multiplicativity of Perverse Filtration for Hilbert Schemes of Fibered Surfaces

TL;DR

This paper proves the multiplicativity of the perverse filtration on certain Hilbert schemes and Hitchin systems, confirming conjectural predictions about their perverse numbers and their relation to the $P=W$ conjecture.

Contribution

It establishes the multiplicativity of the perverse filtration for Hilbert schemes of fibered surfaces and Hitchin systems, and computes their perverse numbers, supporting key conjectures.

Findings

Perverse filtration is multiplicative for Hilbert schemes of fibered surfaces.

Perverse numbers of Hitchin moduli spaces match conjectural predictions.

Supports the $P=W$ conjecture and related conjectures for small n.

Abstract

Let $S\to C$ be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on $H^*(S^{[n]},\mathbb{Q})$ for the natural morphism $S^{[n]}\to C^{(n)}$. We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of $n$ the perverse numbers match the predictions of the numerical version of the de Cataldo-Hausel-Migliorini $P=W$ conjecture and of the conjecture by Hausel, Letellier and Rodriguez-Villegas.