Cohomology of quotients in symplectic and algebraic geometry.

*(English)*Zbl 0553.14020In these notes a general procedure is developed for computing the cohomology of quotients of group actions. If K is a compact Lie group acting symplectically on a symplectic manifold X, the cohomology of the quotient X/K is related to the equivariant cohomology of X, and the equivariant cohomology of X is expressed in terms of the equivariant cohomology of the strata of a stratification of X. This stratification is induced by the norm square of the moment map, as a mildly degenerated Morse function. - In the case of a reductive algebraic group G over an algebraically closed field k, acting on a nonsingular projective k- variety X, the author uses Kempf’s class of optimal destabilizing one parameter subgroups. She rediscovers the stratification introduced by the reviewer in Invent. Math. 55, 141-163 (1979; Zbl 0401.14006) and the corresponding desingularizations. If \(k={\mathbb{C}}\) and G is the complexification of the compact group K, then the two stratifications coincide. Results are presented on the cohomology and the Hodge numbers of X/G. Some detailed examples are given.

Reviewer: W.H.Hesselink

##### MSC:

14L30 | Group actions on varieties or schemes (quotients) |

55N25 | Homology with local coefficients, equivariant cohomology |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14M17 | Homogeneous spaces and generalizations |