Equivariant cobordism for torus actions

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 ΩT*(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 ΩG*(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