Deformations of Chow groups via cyclic homology

TL;DR

This paper investigates infinitesimal deformations of Chow groups for smooth projective varieties over fields of characteristic zero, linking them to Hodge cohomology and extending prior algebraic results.

Contribution

It extends the understanding of Chow group deformations from algebraic over $Q$ fields to all characteristic zero fields, under Hodge cohomology vanishing conditions.

Findings

Isomorphism between deformed Chow groups and Hodge cohomology tensor ideals.

Extension of Bloch's results to arbitrary characteristic zero fields.

Partial affirmative answer to pro-representability linked to Hodge vanishing.

Abstract

Let $X$ be a smooth projective variety over an arbitrary field $k$ of characteristic zero. We explore infinitesimal deformations of the Chow group $CH^{p}(X)$ via its formal completion $\widehat{CH}^{p}$, a functor defined on the category of local augmented Artinian $k$-algebras. Under a natural vanishing condition on Hodge cohomology groups, for certain augmented graded Artinian $k$-algebras $A$ with the maximal ideal $m_{A}$, we prove that \[ \widehat{CH}^{p}(A) \cong H^{p}(X, \Omega^{p-1}_{X/ k})\otimes_{k}m_{A}. \]This extends earlier results of Bloch and others from the case where $k$ is algebraic over $\mathbb{Q}$ to arbitrary fields of characteristic zero,and gives a partial affirmative answer to a general question linking the pro-representability of Chow groups to a specific set of Hodge-theoretic vanishing conditions.