Vector fields, torus actions and equivariant cohomology

Jim Carrell, Kiumars Kaveh and Volker Puppe

Vol. 232 (2007), No. 1, 61–76

DOI: 10.2140/pjm.2007.232.61

Abstract

We give an explicit connection between the holomorphic equivariant cohomology as defined by Carrell and Lieberman and the usual equivariant cohomology of Borel and Cartan.

Let X be a smooth complex projective variety equipped with a ℂ^∗-action with fixed point set Z. By results of Carrell and Lieberman, there exists a filtration F₀ ⊂ F₁ ⊂⋯ of H^∗(Z, ℂ) such that GrH^∗(Z, ℂ)≅H^∗(X, ℂ) as graded algebras. We give here an explicit connection between this filtration and the ℂ^∗-equivariant cohomology of X.

Keywords

equivariant cohomology, holomorphic vector field, equivariant vector field, torus action

Mathematical Subject Classification 2000

Primary: 14F43

Secondary: 57R91

Milestones

Received: 30 March 2006

Revised: 20 October 2006

Accepted: 23 October 2006

Published: 1 September 2007