Oriented Borel-Moore homologies of toric varieties

TL;DR

This paper extends the K"unneth formula to OBM-homology theories for toric varieties, providing tools for computing algebraic cobordism groups of singular toric varieties and establishing a universal coefficient theorem.

Contribution

It generalizes the K"unneth formula to OBM-homology theories for toric varieties and offers a new method to compute algebraic cobordism groups of singular toric varieties.

Findings

Derived a universal coefficient theorem for operational cohomology rings.

Provided a description of homology groups for smooth toric varieties.

Performed explicit computations of algebraic cobordism groups.

Abstract

We generalize the K\"unneth formula for Chow groups to an arbitrary OBM-homology theory satisfying descent (e.g. algebraic cobordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a description for the homology groups of smooth toric varieties, that allows the calculation of algebraic cobordism groups of singular toric varieties. Some computations are carried out.