Rational intersection cohomology of quotient varieties

Abstract

Geometric invariant theory assigns a projective "quotient" variety to any linear action of a complex reductive Lie group G on a complex projective variety X. In this paper a procedure is described for computing the dimensions of the rational intersection homology groups of the quotient when X is nonsingular (see Theorem 3.1). The only condition which the action must satisfy is that the set of stable points for the action is not empty.The basic idea goes like this. Let the geometric invariant theory quotient of X by G be denoted by X//G to distinguish it from the ordinary topological quotient space X/G. In 1-12] a formula was given for the Betti numbers of the quotient in the special case when every semistable point is (properly) stable. In this case the quotient is everywhere locally isomorphic to the quotient of a nonsingular variety by a finite group action and hence its rational intersection cohomology groups are isomorphic to its ordinary rational cohomology groups. On the other hand given any linear action of G on X such that the set of stable points is nonempty, there is a systematic way to blow up X along a sequence of nonsingular subvarieties to obtain a variety X with a linear action of G lifting the action on X such that every semistable point of ~" is stable (see [13]). The quotient X//G may be thought of as a partial desingularisation of X//G (the more serious singularities have been resolved) and its Betti numbers can be computed by the method described in [12]. Finally in this paper it is shown how to relate the intersection Betti numbers of X//G to the Betti numbers of X//G (Theorem 3.1). Thus we end up with a procedure for computing the dimensions of the rational intersection cohomology groups of the quotient X//G in terms of the rational cohomology of certain nonsingular linear sections Z of X and the rational cohomology of the classifying spaces of certain reductive subgroups H of G, provided that we know how no(H ) acts on H*(Z) for appropriate Z and H acting on Z.The formulas for these intersection Betti numbers can become very complicated and involve many terms, especially as the rank of G increases. How-1 Current address: Mathematical Institute, Oxford, UK

Cohomology pairings on singular quotients in geometric invariant theory

“…This map κ ss M : H * K (M ss ) → IH * (M//G) is surjective; the proof of this in [27] is flawed but an alternative proof is given in [41].…”

Towards Non-Reductive Geometric Invariant Theory

“…This partial desingularization X//H (which has only orbifold singularities) is itself a projective completion of the geometric quotient X s /H. It can be represented as X//H =Ỹ //G whereỸ ss =Ỹ s is obtained from Y ss by blowing up along its intersection with Z and removing the proper transform of the subvariety consisting of those (w, x) ∈ P 2 × P n with x representing n points on P 1 exactly half of which coincide at some u = [a : b] ∈ P 1 , and with w = [1 : ta : tb] for some t ∈ C. We obtain P t ( X//H) = P For more details see [37]. The Poincaré polynomial of the geometric quotient X s /H when n is even is given by…”

Intersection cohomology of quotients of nonsingular varieties

“…It is proved in [Kir86b] Lemma 2.9 that N / /K is homeomorphic to a neighborhood of X (h) = GM ss H / /G and hence N U / /K is homeomorphic to a neighborhood of U ∩ X (h) . We identify N U with a tubular neighborhood of GM ss H ∩ φ −1 (U ) and identify N U / /K with a neighborhood…”