On the cohomology ring of the moduli space of rank 2 vector bundles on a curve

Abstract

Let N g be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus g. This paper gives a complete and very simple description of the rational cohomology ring H ∗( N g) . A structural formula is proved for H ∗( N g) , which was originally conjectured by Mumford. It is shown that the first relation in genus g between the standard generators satisfies a recurrence relation, first found by Zagier, and that the invariant subring for the mapping class group is a complete intersection ring. A Gröbner basis is found for the ideal of invariant relations; this leads to a natural monomial basis for H ∗( N g) .