Deformations of Chow groups via cyclic homology
Sen Yang
TL;DR
The paper studies infinitesimal deformations of the Chow groups CH^{p}(X) over a field k of characteristic zero by examining the formal completion \\widehat{CH}^{p} on local augmented Artinian k-algebras. It leverages cyclic and Hochschild homology, via Goodwillie’s perspective and the SBI exact sequence, to relate deformations to relative cyclic homology sheaves and then to Hodge cohomology. Under a specific Hodge-vanishing condition, it proves the canonical isomorphism \\widehat{CH}^{p}(A) \\cong H^{p}(X, \\Omega^{p-1}_{X/k}) \\otimes_{k} m_{A} for graded Artinian algebras with A_{0}=k, generalizing Bloch’s results from fields algebraic over \\mathbb{Q} to arbitrary characteristic zero fields. This yields a partial affirmative answer to linking pro-representability of Chow groups with Hodge theory and offers a framework connecting algebraic cycles to cyclic homology via relative theories.
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.