Perverse filtrations, Hilbert schemes, and the P=W conjecture for parabolic Higgs bundles

TL;DR

This paper proves the P=W conjecture for parabolic Higgs bundles of arbitrary rank in specific affine Dynkin types, using tautological classes on Hilbert schemes of points on elliptic surfaces.

Contribution

It establishes the P=W conjecture for a broad class of parabolic Higgs bundles in new affine Dynkin types, expanding previous results.

Findings

Proves P=W conjecture for affine Dynkin types $ ilde{A}_0$, $ ilde{D}_4$, $ ilde{E}_6$, $ ilde{E}_7$, $ ilde{E}_8$.

Uses tautological classes on Hilbert schemes of points on elliptic surfaces.

Connects perverse filtrations with the geometry of parabolic Higgs bundles.

Abstract

We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $\tilde{A}_0$, $\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$. Our proof relies on the study of the tautological classes on the Hilbert scheme of points on an elliptic surface with respect to the perverse filtration.