Algebraic cobordism theory attached to algebraic equivalence

TL;DR

This paper develops a new algebraic cobordism theory modulo algebraic equivalence, connecting it with existing theories like Chow groups and semi-topological K-groups, and computes it for certain varieties.

Contribution

It introduces a cobordism theory modulo algebraic equivalence that generalizes and unifies several existing theories, with new computational results.

Findings

Reproduces Chow groups modulo algebraic equivalence

Aligns with algebraic cobordism with finite coefficients

Provides computations for low-dimensional varieties

Abstract

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.