Characteristic Classes of Stable Bundles of Rank 2 over an Algebraic Curve

Let X be a complete nonsingular algebraic curve over C and L a line bundle of degree 1 over X. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant L over X form a nonsingular projective variety S(X). The Betti numbers of S(X) are also known. In this paper we define certain distinguished cohomology classes of S(X) and prove that these classes generate the rational cohomology ring. We also obtain expressions for the Chern character and Pontrjagin classes of S(X) in terms of these generators. Introduction. Let X be a complete nonsingular algebraic curve of genus g > 2 over the complex numbers and let L be a line bundle of degree 1 over X. The isomorphism classes of stable bundles of rank 2 and determinant L over X form a nonsingular projective variety S(X) (see [2], [6]), whose Betti numbers were calculated in [3]. The main object of this paper is to show that certain naturally occurring elements generate the rational cohomology ring of S(X) (Theorem 1). We shall also obtain expressions for the Chern character and Pontrjagin classes of S(X) in terms of these generators. My thanks are due to S. Ramanan for informing me of his work on this topic in advance of publication (see [5]). He has obtained generators and relations for the rational cohomology ring of S(X) in the case g = 3. He has also obtained some information for the spaces of stable bundles of rank « (in particular a generalisation of Corollary 1 to Theorem 2) by methods similar to those used in the proof of Proposition 2.2. Unless the contrary is indicated, all cohomology groups in this paper will have integral coefficients. Also, if E is any bundle over V x W (or W x V) and v £ V, we shall denote by E the bundle over W obtained by restricting E to \v\ x W (or W x\v\). 1. Statement of the main theorem. We recall [1, §l] that there exists an algebraic vector bundle U over S(X) x X with the property that, for all s £ S(X), Received by the editors September 9, 1971. AMS 1970 subject classifications. Primary 14D20, 14F05, 14F25; Secondary 55F40, 57D20.