Rational intersection cohomology of quotient varieties. II

Abstract

Given a linear action of a complex reductive group G on a complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory [18]. From the construction of this quotient it is not immediately obvious how its topology is related to the topology of X and the group action. However, when X is nonsingular it is possible (at least if one technical condition is satisfied) to compute the intersection Betti numbers of X//G in terms of the rational cohomology of certain nonsingular linear sections Z of X and the classifying spaces of certain reductive subgroups H of G (see [15]). The technical condition is that there should exist at least one point of X which is (properly) stable for the action of G: this is satisfied in most interesting examples such as those coming from moduli problems. The sections of X and subgroups of G which occur have a simple description in terms of the representation of G which induces the linear action on X. It thus seems reasonable to ask what happens when X is singular. The aim of this paper is to show that the procedure for computing the intersection Betti numbers of X//G can be generalised to the singular case (see (2.25) and (2.28)). However, now one needs to know not only the intersection cohomology of the various sections Z of X but also how they are positioned relative to the singularities of X. In the special case when they meet the singularities transversely then one has exactly the same formulas as when X is nonsingular except that ordinary cohomology is replaced by intersection cohomology throughout. The situation which it is most important to understand is that in which every semistable point is stable, and this case is considered in detail in w 2 after the necessary preliminaries have been covered in w 1. Some brief remarks are made about the more general case when one only assumes that there exist some stable points. A simple example is worked out (2.27), which shows that the formulas can also be used in reverse to compute the intersection Betti numbers of X from knowledge of the intersection Betti numbers of the quotient X//G (even simply from the knowledge that these vanish in dimensions greater than dim~X//G).