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The geometry and topology of quotient varieties of torus actions. (English) Zbl 0812.14031
The author studies the topology of the quotient variety of a complex algebraic projective variety $$X$$ with an action of a complex algebraic torus $$(\mathbb{C}^*)^ n$$. As a main result, he obtains an inductive formula for the intersection Betti numbers. The formula includes singular quotients. One can always find a rationally nonsingular quotient and a canonical map that is a small resolution in the sense of Goresky- MacPherson. The most important quotients under consideration are those that can be understood as symplectic reduced spaces.
The author starts with a nice introduction to this subject. The main part of the paper contains an adequate stratification, the proof that there exist small resolutions and the decomposition theorem for the intersection homology in the symplectic case. In a further step this is done analogously in the so-called semigeometric case which generalizes the symplectic one. But the results are smaller: Vanishing property of intersection homology in odd degree and isomorphism to rational intersection groups are transferable from the fixed-point set to the quotient. Finally, an application to flag varieties is given. See also F. C. Kirwan, “Cohomology of quotients in symplectic and algebraic geometry”, Math. Notes 31 (1984; Zbl 0553.14020).