The motive moduli spaces of rank two vector bundles over a curve

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 Poincar\'e-Hodge polynomial. When the degree is even the moduli space is a singular projective variety, we compute the pure motivic Poincar\'e polynomial and show that only two weights can occur in each cohomology group. As corollaries we obtain the isogeny type of some intermediate jacobians of the smooth moduli space and the motive and Hodge numbers of Seshadri's smooth model for the singular moduli space.