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

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.

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

### MathOverflow Questions:

In GIT, why are the semistable/unstable loci defined pointwise, instead of defining semistable/unstable subschemes?### 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 |