Seminar on Arithmetic Geometry

Charanya Ravi (Indian Statistical Institute, Bangalore Centre)

The classical Atiyah-Bott localization theorem in equivariant singular cohomology is one of the primary computational tools in enumerative geometry when the relevant parameter space has a natural torus action. The theorem asserts that the equivariant cohomology of a space with group action can be recovered, up to inversion of some elements, from the equivariant cohomology of the fixed point subspace. To understand various moduli problems, there is a need to access general parameter spaces (singular and stacky) and to produce refined counts (in different cohomology theories).

This talk will first discuss a unified Atiyah-Bott localization theorem for equivariant Borel-Moore homology theories of algebraic stacks. We will then discuss a categorified version of the result which allows us to deduce the theorem for all oriented theories (cohomology and Borel-Moore homology). The talk is based on joint works with Dhyan Aranha, Adeel Khan, Alyosha Latyntsev, and Hyeonjun Park.

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