The motive of some moduli spaces of vector bundles over a curve

TL;DR

This paper investigates the motive and cohomological properties of moduli spaces of semistable rank two vector bundles over a curve, revealing their Hodge structures, Euler characteristics, and intermediate Jacobians.

Contribution

It computes the absolute Hodge motive and Hodge-Poincare polynomial for odd degree cases and analyzes the cohomology and Euler characteristics for even degree cases.

Findings

Computed the absolute Hodge motive for odd degree moduli spaces.

Determined pure Euler characteristics and weight structures for even degree moduli spaces.

Identified the isogeny types of certain intermediate Jacobians.

Abstract

We study the motive of the moduli spaces of semistable rank two vector bundles over an algebraic curve. When the degree is odd the moduli space is a smooth projective variety, we obtain the absolute Hodge motive of this, and in particular the Hodge-Poincare polynomial. When the degree is even the moduli space is a singular projective variety, we compute pure Euler characteristics and show that only two weights can occur in each cohomology group, we also see that its cohomology is pure up to a certain degree. As a by-product we obtain the isogeny type of some intermediate jacobians of the moduli spaces.