Equivariant intersection cohomology.

*(English)*Zbl 0803.55002
Deodhar, Vinay (ed.), Kazhdan-Lusztig theory and related topics. Proceedings of an AMS special session, held May 19-20, 1989 at the University of Chicago, Lake Shore Campus, Chicago, IL, USA. Providence, RI: American Mathematical Society. Contemp. Math. 139, 5-32 (1992).

In this paper (which was first circulated as an IHES preprint in 1986) we find the first public appearance of the intersection cohomology by means of differential forms; this version of the intersection cohomology is also due to Goresky and MacPherson.

The objective of this work is the introduction of the notion of equivariant intersection cohomology to study the action of compact groups on singular varieties (see [R. Joshua, Math. Z. 195, 239-253 (1987; Zbl 0637.14014)] for another point of view). For a smooth torus action \(\Phi: T\times X\to X\), and using the above presentation of the intersection cohomology, the author gives a perverse version of the Cartan equivariant complex which hypercohomology computes the equivariant intersection cohomology \(IH^*_{p,T} (X)\). He shows that this cohomology satisfies Poincaré duality, taking as base ring the cohomology ring \(H^* (BT)\). Notice that considering \(X\) as a compact manifold this result gives a new Poincaré duality for equivariant cohomology. The work ends with a localization theorem and an application to the study of fixed point set of tori actions on projective singular varieties.

For the entire collection see [Zbl 0784.00017].

The objective of this work is the introduction of the notion of equivariant intersection cohomology to study the action of compact groups on singular varieties (see [R. Joshua, Math. Z. 195, 239-253 (1987; Zbl 0637.14014)] for another point of view). For a smooth torus action \(\Phi: T\times X\to X\), and using the above presentation of the intersection cohomology, the author gives a perverse version of the Cartan equivariant complex which hypercohomology computes the equivariant intersection cohomology \(IH^*_{p,T} (X)\). He shows that this cohomology satisfies Poincaré duality, taking as base ring the cohomology ring \(H^* (BT)\). Notice that considering \(X\) as a compact manifold this result gives a new Poincaré duality for equivariant cohomology. The work ends with a localization theorem and an application to the study of fixed point set of tori actions on projective singular varieties.

For the entire collection see [Zbl 0784.00017].

Reviewer: M.Saralegi-Aranguren (Madrid)