Pro-representability of Chow groups and Hodge numbers

Let $k$ be an algebraic field extension of $\mathbb{Q}$ and let $X$ be a smooth projective variety over $k$ of dimension $d\ge 2$. We study the pro-representability of the Chow group $C{H}^p(X)$ with $2\le p\le d$. When certain Hodge numbers of $X$ vanish, namely, $H^p(X,{\mathrm{\Omega }}_{X\mathrm{/}k}^i)=H^{p+1}(X,{\mathrm{\Omega }}_{X\mathrm{/}k}^i)=\cdots =H^{2p-1-i}(X,{\mathrm{\Omega }}_{X\mathrm{/}k}^i)=0$ for $i$ such that $0\le i\le p-2$, we prove that the formal completion ${\widehat{CH}}^p(A)$ of $C{H}^p(X)$ at a local augmented Artinian $k$-algebra $A$ with the maximal ideal $m_A$ satisfies $${\widehat{CH}}^p(A)\cong H^p(X,{\mathrm{\Omega }}_{X\mathrm{/}k}^{p-1}){\otimes }_k{m}_A\mathrm{.}$$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 $X$.