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**The topology of quotient varieties.**
*(English)*
Zbl 0692.14032

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.

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.

### MSC:

14M17 | Homogeneous spaces and generalizations |

14F45 | Topological properties in algebraic geometry |

14L30 | Group actions on varieties or schemes (quotients) |