Deformations of Chow groups via cyclic homology

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 \[ \widehat{CH}^{p}(A) \cong H^{p}(X, Ω^{p-1}_{X/ k})\otimes_{k}m_{A}. \]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.