On the Cohomology of Moduli Spaces of Rank Two Vector Bundles Over Curves

Abstract

Let C be a Riemann surface, L a line bundle over C, and n a natural number. Then there is a moduli space of stable n-dimensional vector bundles E over C with determinant bundle Λⁿ(E) ≡ L; this moduli space is smooth but in general non-compact and can be compactified by the suitable addition of semi-stable bundles to a projective, but in general singular, variety N _c,n,L .The topology of this variety depends only on the genus g of C and the degree d of L (in fact, only on d modulo n, since tensoring E with a fixed line bundle L ₁ replaces L by L ⊗ L ⁿ₁ ), so we will also use the notation N _g,n,d . We will be studying only the case n = 2, and hence will drop the n and replace d by ε = (−1)^d in the notation. Thus for each g we have two moduli spaces of stable 2-dimensional bundles N ⁻_g and N ⁺_g , both projective varieties of complex dimension 3g − 3. We will be looking mostly at the smooth space N ⁻_g and will often denote it simply N _g .