Motives of moduli spaces of bundles on curves via variation of stability and flips

TL;DR

This paper investigates the motives of moduli spaces of vector bundles on curves, analyzing how these motives change under stability condition variations and establishing motivic analogues of classical theorems.

Contribution

It provides explicit descriptions of motive variations during wall-crossings and proves a motivic version of Harder-Narasimhan's theorem for moduli spaces.

Findings

Motivic invariance under wall-crossings for parabolic Higgs bundles.

Explicit formulas for motives of rank 2 vector bundles and Higgs bundles.

Motivic relations between moduli spaces with and without fixed determinants.

Abstract

We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall-crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall-crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulas for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).