The Chow ring of the Classifying Space BSO(2n,C)

We compute the Chow ring of the classifying space BSO(2n, C) in the sense of Totaro using the fibration Gl(2n)/SO(2n) → BSO(2n) → BGl(2n) and a computation of the Chow ring of Gl(2n)/SO(2n) in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to 2 times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents 1/2(n − 1)! times the n Chern class of the representation of SO(2n) whose highest weight vector is twice that of the halfspin representation. Throughout this paper we write Gl(n), O(n), SO(n), etc. for the the complex algebraic groups of these types. They are homotopy equivalent to the compact groups U(n), O(n,R), and SO(n,R). This paper is devoted to computing the Chow ring of the classifying space BSO(2n). The definition of the Chow ring of a classifying space we use is that of Totaro [14]. This is defined as the limit of Chow rings of finite dimensional algebraic varieties approximating the classifying space, and coincides with the ring of characteristic classes for principal SO(2n,C) bundles over smooth varieties. Our main result is: