Kirwan, Frances Clare Cohomology of quotients in symplectic and algebraic geometry. (English) Zbl 0553.14020 Mathematical Notes, 31. Princeton, New Jersey: Princeton University Press. III, 210 p. $ 17.50 (1984). In 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 Cited in 35 ReviewsCited in 280 Documents MathOverflow Questions: Quotients of Kähler manifolds 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 Keywords:cohomology of quotients of group actions; equivariant cohomology; stratification; Morse function; Hodge numbers Citations:Zbl 0415.14029; Zbl 0401.14006 PDF BibTeX XML OpenURL