Equivariant Cobordism for Torus Actions

TL;DR

This paper develops the theory of equivariant cobordism for schemes with torus actions, establishing relations with ordinary cobordism, structure theorems, and localization results, with implications for smooth projective varieties.

Contribution

It provides explicit relations between equivariant and ordinary cobordism for schemes with torus actions and extends results to actions of connected linear algebraic groups.

Findings

Established localization theorems in equivariant cobordism

Derived structure theorems for cobordism rings

Described cobordism rings of certain smooth projective varieties

Abstract

We study the equivariant cobordism theory of schemes for torus actions. We give the explicit relation between the equivariant and the ordinary cobordism of schemes with torus action. We deduce analogous results for action of arbitrary connected linear algebraic groups. We prove some structure theorems for the equivariant and ordinary cobordism of schemes with torus action and derive important consequences. We establish the localization theorems in this setting. These are used to describe the structure of the ordinary cobordism ring of certain smooth projective varieties.