Bivariant Algebraic Cobordism

TL;DR

This paper develops a bivariant theory for algebraic cobordism, leading to new operational cobordism rings for schemes and G-schemes, with explicit descriptions for toric varieties.

Contribution

It introduces a bivariant framework for algebraic cobordism and describes operational cobordism rings, including explicit formulas for toric varieties.

Findings

Operational cobordism rings are constructed for schemes and G-schemes.

For toric varieties, the T-equivariant cobordism ring is described as piecewise graded power series.

The framework generalizes algebraic cobordism to a bivariant setting.

Abstract

We associate a bivariant theory to any suitable oriented Borel-Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.