Pro-representability of Chow groups and Hodge numbers
Sen Yang
TL;DR
This work addresses when the infinitesimal deformations of Chow groups CH^{p}(X) are governed by a pro-representable functor. By leveraging Bloch's formula CH^{p}(X) \\otimes \\mathbb{Q} \\cong H^{p}(X, K^{M}_{p}(O_{X})) \\otimes \\mathbb{Q} and a de Rham–cyclic homology bridge, the authors derive a cohomological criterion: if certain Hodge groups H^{p}(X,\\Omega^{i}_{X/k}) vanish in a specified range, then the formal completion \\widehat{CH}^{p}(A) is isomorphic to H^{p}(X,\\Omega^{p-1}_{X/k}) \\otimes_{k} m_{A}, for A in Art_{k}. This yields pro-representability of \\widehat{CH}^{p} and unifies prior results for p=2 and p=3 under a single Hodge-vanishing condition. The approach connects the deformation theory of algebraic cycles with the Hodge structure of X through de Rham and cyclic homology, offering a practical criterion and illuminating the role of Hodge numbers in the pro-representation problem.
Abstract
Let $k$ be an algebraic field extension of $\mathbb{Q}$ and let $X$ be a smooth projective variety over $k$ of dimension $d \geq 2$. We study the pro-representability of the Chow group $CH^{p}(X)$ with $2 \leq p \leq d$. When certain Hodge numbers of $X$ vanish, namely, $H^{p}(X,Ω^{i}_{X/k})=H^{p+1}(X,Ω^{i}_{X/k})= \cdots =H^{2p-1-i}(X,Ω^{i}_{X/k})=0$ for $i$ such that $0 \leq i \leq p-2$, we prove that the formal completion $\widehat{CH}^{p}(A)$ of $CH^{p}(X)$ at a local augmented Artinian $k$-algebra $A$ with the maximal ideal $m_{A}$ satisfies \[ \widehat{CH}^{p}(A) \cong H^{p}(X, Ω^{p-1}_{X/ k})\otimes_{k}m_{A}. \]This provides a unified cohomological criterion for the pro-representability of the functor $\widehat{CH}^{p}$, generalizing earlier work by Bloch, Stienstra, and Mackall for $p=2$ and $p=3$. Our result reveals an intrinsic connection between the deformation theory of algebraic cycles and the Hodge structure of $X$.