Pro-representability of Chow groups and Hodge numbers

TL;DR

This paper establishes a cohomological criterion for the pro-representability of Chow groups of smooth projective varieties, linking deformation theory of algebraic cycles to Hodge structures under certain vanishing conditions.

Contribution

It generalizes previous results by providing a unified criterion for pro-representability of Chow groups based on Hodge number vanishing conditions.

Findings

Proves isomorphism between formal completion of Chow groups and Hodge cohomology tensor with maximal ideal.

Provides a criterion that unifies and extends earlier work for p=2 and p=3.

Reveals a deep connection between deformation theory of cycles and Hodge structures.

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, $H^{p}(X,\Omega^{i}_{X/k})=H^{p+1}(X,\Omega^{i}_{X/k})= \cdots =H^{2p-1-i}(X,\Omega^{i}_{X/k})=0$ 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 \[ \widehat{CH}^{p}(A) \cong H^{p}(X, \Omega^{p-1}_{X/ k})\otimes_{k}m_{A}. \]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 $X$.