Partial desingularisations of quotients of nonsingular varieties and their Betti numbers (Q1075388)

Let G be a reductive group acting linearly on a nonsingular complex projective variety X and let X//G denote the quotient according to Mumford's geometric invariant theory. If every semistable point is stable, then a formula was previously obtained by the author for the rational cohomology of X//G [''Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020)]. This paper treats the general case, showing that there is a systematic way of blowing up X along a sequence of nonsingular G-invariant subvarieties to obtain a variety \(\tilde X\) with a linear action of G such that every semistable point of \(\tilde X\) is stable (assuming that the set of stable points in X is nonempty). Then \(\tilde X/\)/G is a partial desingularisation of X//G and there is a formula for the rational cohomology of \(\tilde X/\)/G in terms of the cohomoloy of X and certain linear sections of X, together with the cohomoloy of the classifying spaces of G and some reductive subgroups of G. - Some examples including moduli spaces of hyperelliptic curves are given in the end.