Cohomology theories and rings of functions

Abstract: Extending the classical Poincare-Hopf theorem, the work of Akyildiz, Carrell, Liebermann, Sommese shows how to recover the cohomology ring of a smooth projective variety from isolated zeros of a vector field. Thirty years later, Brion and Carrell showed how to find the spectrum of the torus-equivariant cohomology as a geometrically defined scheme, provided that the Borel of SL_2 acts with a single fixed point of the regular unipotent. In a joint work with Tamas Hausel we demonstrate how to see the spectrum of G-equivariant cohomology, if G is a linear group acting with similar assumptions. This condition covers many interesting cases, including flag varieties and Bott–Samelson resolutions. I will present this work and also show how to see the equivariant cohomology rings of spherical varieties as rings of functions on non-affine schemes. Besides, there are a lot of new directions and open questions I would like to advertise. This in particular concerns general, potentially singular varieties, as well as other equivariant cohomology theories.

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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