Equivariant cobordism for torus actions

Abstract

We study the equivariant cobordism groups for the action of a split torus T on varieties over a field k of characteristic zero. We show that for T acting on a variety X, there is an isomorphism Ω^T_*(X)⊗_Ω*(BT) L →^≅ Ω_*(X) source. As applications, we show that for a connected linear algebraic group G acting on a k-variety X, the forgetful map Ω^G_*(X) → Ω_*(X) is surjective with rational coefficients. As a consequence, we describe the rational algebraic cobordism ring of algebraic groups and flag varieties. We prove a structure theorem for the equivariant cobordism of smooth projective varieties with torus action. Using this, we prove various localization theorems and a form of Bott residue formula for such varieties. As an application, we show that the equivariant cobordism of a smooth variety X with torus action is generated by the invariant cobordism cycles in Ω_*(X) as Ω^∗(BT)-module.