Motivic Steenrod problem away from the characteristic

TL;DR

This paper explores an algebraic geometry analogue of the Steenrod problem, demonstrating that not all motivic cohomology classes of singular varieties are pushforwards of fundamental classes, contrasting with the smooth case.

Contribution

It introduces the motivic Steenrod problem in algebraic geometry and provides examples showing the failure of pushforward representation for singular varieties.

Findings

Not all motivic cohomology classes of singular schemes are pushforwards.

The Chow ring of a singular variety cannot be obtained as a quotient of its algebraic cobordism ring.

Resolution of singularities suffices for smooth varieties but not for singular ones.

Abstract

In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in algebraic geometry: is every element on the Chow line of the motivic cohomology of $X$ the pushforward of a fundamental class along a projective derived-lci morphism? If $X$ is a smooth variety over a field of characteristic $p \geq 0$, then a positive answer to this question follows up to $p$-torsion from resolution of singularities by alterations. However, if $X$ is singular, then this is no longer necessarily so: we give examples of motivic cohomology classes of a singular scheme $X$ that are not $p$-torsion and are not expressible as such pushforwards. A consequence of our result is that the Chow ring of a singular variety cannot be expressed as a quotient of its algebraic cobordism ring, as suggested by the first-named-author in his thesis.