Equivariant cobordism for torus actions

We study the equivariant cobordism groups for the action of a split torus TT on varieties over a field kk of characteristic zero. We show that for TT acting on a variety XX, there is an isomorphism Ω∗T(X)⊗Ω∗(BT)L⟶≅Ω∗(X). As applications, we show that for a connected linear algebraic group GG acting on a kk-variety XX, 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 XX with torus action is generated by the invariant cobordism cycles in Ω∗(X)Ω∗(X) as Ω∗(BT)Ω∗(BT)-module.