Smoothing low-dimensional cycles in algebraic cobordism

TL;DR

This paper proves that under certain dimension conditions, every cycle in algebraic cobordism can be expressed as a combination of smooth subvariety cycles, extending previous Chow group results.

Contribution

It generalizes Kollár and Voisin's smoothability result from Chow groups to algebraic cobordism groups for smooth projective varieties.

Findings

Every cycle in $ ext{Omega}_d(X)$ is smoothable when $2d< ext{dim}(X)$.

Cycles can be expressed as linear combinations of smooth subvariety cycles.

The result extends known Chow group smoothability to algebraic cobordism.

Abstract

We show that every cycle in the degree $d$ algebraic cobordism group $\Omega_d(X)$ of a smooth projective variety $X$ over a field of characteristic $0$ is smoothable when $2d<\dim(X)$, that is, it can be written as a linear combination of cycles represented by smooth closed subvarieties of $X$. This generalizes a result of Koll\'ar and Voisin from Chow groups to algebraic cobordism groups.