The topology of quotient varieties (Q583315)

Suppose X is a scheme acted on by a reductive algebraic group G, and suppose a space X/G exists, together with an affine morphism \(\pi\) : \(X\to X/G\), and X/G can be covered by open affines such that over them \(\pi\) is given by the map \(Spec(A)\to Spec(A^ G)\). Then we call X/G the quotient of X by G. What we try to do here is to investigate the relation between the ordinary, complex topology of X and that of X/G. The key results of part 1 of the paper are that there exists a closed subset \(C\subset Spec(A)\) such that (a) The composite map \(C\to Spec(A)\to Spec(A^ G)\) is proper and surjective. (b) C is a deformation retract of Spec(A), with a deformation retraction that commutes with \(\pi:\quad Spec(A)\to Spec(A^ G).\) Part 2 concerns itself with the study of the Chern and Pontrjagin classes of bundles on the quotient: Sections 7, 8 and 9 are largely the technical background needed for section 10, where a method to show how to prove vanishing for Chern or Pontrjagin rings on quotient varieties is indicated. In sections 11 and 12 this technique, called the ``program'', is applied to one example. We obtain partial results on a conjecture of Ramanan about the vanishing of Pontrjagin classes on the moduli space of stable vector bundles of rank 2 and degree 1 over an algebraic curve. In the sequel, there are outlined some rather wild conjetures.