Intermediate Jacobians and Hodge Structures of Moduli Spaces Donu Arapura and Pramathanath Sastry

We work throughout over the complex numbers C, i.e. all schemes are over C and all maps of schemes are maps of C-schemes. A curve, unless otherwise stated, is a smooth complete curve. Points mean geometric points. We will, as is usual in such situations, toggle between the algebraic and analytic categories without warning. For a curve X , SUX(n, L) will denote the moduli space of semi-stable vector bundles of rank n and determinant L. The smooth open subvariety defining the stable locus will be denoted SU X(n, L). We assume familiarity with the basic facts about such a moduli space as laid out, for example in [22, pp. 51–52,VI.A] (see also Theorems 10, 17 and 18 of loc.cit.). When L is a line bundle of degree coprime to n, the moduli spaces SUX(n,L) and SU X(n,L) coincide, and are therefore smooth and projective. The cohomology groups H(SUX(n,L),Q) carry pure Hodge structures which can, in principle, be determined by using a natural set of generators (Atiyah-Bott [2]) and relations (Jeffrey-Kirwan [13]) for the cohomology ring; we will say more about this later. When the degree of L is not coprime to n and g > 2, the situation is complicated by the fact that SUX(n,L) is singular and SU s X(n,L) nonprojective. Thus the cohomology groups of these spaces carry (a-priori) mixed Hodge structures, and it are these structures that we wish to understand. Our main results concerns the situation in low degrees.