Algebraic cycles and Hitchin systems

TL;DR

This paper advances the understanding of the motivic structure of Hitchin systems for GL_n, proving conjectures related to algebraic cycles, filtrations, and symmetries, with implications for algebraic geometry and representation theory.

Contribution

It proves the motivic decomposition conjecture and the relative Lefschetz standard conjecture for Hitchin systems, establishing new links between algebraic cycles and geometric structures.

Findings

Proved the motivic decomposition conjecture for Hitchin systems.

Confirmed the relative Lefschetz standard conjecture in this context.

Established a $oldsymbol{ ext{chi}}$-independence result for Hitchin system motives.

Abstract

The purpose of this paper is to study motivic aspects of the Hitchin system for $\mathrm{GL}_n$. Our results include the following. (a) We prove the motivic decomposition conjecture of Corti-Hanamura for the Hitchin system; in particular, the decomposition theorem associated with the Hitchin system is induced by algebraic cycles. This yields an unconditional construction of the motivic perverse filtration for the Hitchin system, which lifts the cohomological/sheaf-theoretic perverse filtration. (b) We prove that the inverse of the relative Hard Lefschetz symmetry is induced by a relative algebraic correspondence, confirming the relative Lefschetz standard conjecture for the Hitchin system. (c) We show a strong perversity bound for the normalized Chern classes of a universal bundle with respect to the motivic perverse filtration; this specializes to the sheaf-theoretic result obtained earlier by Maulik-Shen. (d) We prove a $\chi$-independence result for the relative Chow motives associated with Hitchin systems. Our methods combine Fourier transforms for compactified Jacobian fibrations associated with integral locally planar curves, nearby and vanishing cycle techniques, and a Springer-theoretic interpretation of parabolic Hitchin moduli spaces.