Pro-representability of Chow groups and Hodge numbers
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917180005679104 |
|---|---|
| author | Yang, Sen |
| author_facet | Yang, Sen |
| contents | 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 $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00390 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pro-representability of Chow groups and Hodge numbers Yang, Sen Algebraic Geometry 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 $X$. |
| title | Pro-representability of Chow groups and Hodge numbers |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2601.00390 |