Partial Desingularisations of Quotients of Nonsingular Varieties and their Betti Numbers

Abstract

When a reductive group G acts linearly on a nonsingular complex projective variety X one can define a projective variety X//G using Mumford's geometric invariant theory. If the condition that every semistable point be (properly) stable is satisfied, this quotient is the ordinary topological quotient of an open subset Xss of the variety by the group. In [K] a formula is obtained for the rational cohomology of X//G under this condition. The formula involves the rational cohomology of X and various linear sections of X, together with the rational cohomology of the classifying spaces of G and certain reductive subgroups of G. In many interesting examples the condition required in [K] is not satisfied. Thus the question arises as to what information we can obtain in general. The quotient X//G can now have serious singularities in contrast to the good case considered in [K] where the only singularities are those caused by finite isotropy groups. It will be shown in this paper that there is a systematic way of blowing up X along a sequence of nonsingular subvarieties to obtain a variety X with a linear action of G such that every semistable point of X is stable. The only assumption that we have to make is that there exists at least one stable point of X. Then X//G is almost a resolution of singularities of X1/G, in the sense that the most serious singularities have been resolved. Moreover there is a formula for the rational cohomology of X//G again involving the rational cohomology of X and certain linear sections of X, together with the rational cohomology of the classifying spaces of G and some reductive subgroups of G (see Theorem 8.14). For convenience we shall assume throughout that G is connected. However the construction of X works in general, and it is straightforward to modify the cohomological formulas to apply to the general case. The construction of X//G can also be modified to apply in some cases when X has no stable points. The layout of the paper is as follows. Section 1 is a review of the basic facts of geometric invariant theory which will be needed and Section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map. In Section 3 semistability and stability in a blow-up of X along a nonsingular C-invariant subvariety are related to semistability and stability in X,