Partial desingularisations of quotients of nonsingular varieties and their Betti numbers. (English) Zbl 0592.14011

Kirwan, Frances Clare

Ann. Math. (2) 122, 41-85 (1985).

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.

Reviewer: A.Dimca

MSC:

14E15	Global theory and resolution of singularities (algebro-geometric aspects)
14L30	Group actions on varieties or schemes (quotients)
14L24	Geometric invariant theory
14D20	Algebraic moduli problems, moduli of vector bundles
14F25	Classical real and complex (co)homology in algebraic geometry

Keywords:

geometric invariant theory; blowing up; partial desingularisation; rational cohomology; moduli spaces of hyperelliptic curves

Citations:

Zbl 0553.14020