On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909986901196800 |
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| author | Yu, Jiahong |
| author_facet | Yu, Jiahong |
| contents | Let $C$ be a completely algebraic closed non-archimedean field over $\mathbb{Q}_p$ and $α,r$ be two positive integers. Denote by $B_α$ the ring $\mathbb{B}_{\mathrm{dR}}^+(C)/(\kerθ)^α$. This paper first constructs a sheafified $p$-adic Riemann--Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on $X_{\mathrm{\acute{e}t}}$, \[ R^1ν_*\big( \mathrm{GL}_r(\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α) \big) \cong \mathrm{MIC}_{r}(X)\{-1\}, \] where the first term is identified with the sheaf of isomorphism classes of $v$-vector bundles with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α$, and the second term is defined as the sheaf of isomorphism classes of integrable connections of rank $r$.
We then define the moduli space of integrable connections on $X$ and the moduli space of $v$-vector bundles on $X$ with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α$, and prove that they are small $v$-stacks in the sense of Scholze.
These constructions generalize Heuer's work on $p$-adic Simpson correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence Yu, Jiahong Algebraic Geometry Let $C$ be a completely algebraic closed non-archimedean field over $\mathbb{Q}_p$ and $α,r$ be two positive integers. Denote by $B_α$ the ring $\mathbb{B}_{\mathrm{dR}}^+(C)/(\kerθ)^α$. This paper first constructs a sheafified $p$-adic Riemann--Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on $X_{\mathrm{\acute{e}t}}$, \[ R^1ν_*\big( \mathrm{GL}_r(\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α) \big) \cong \mathrm{MIC}_{r}(X)\{-1\}, \] where the first term is identified with the sheaf of isomorphism classes of $v$-vector bundles with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α$, and the second term is defined as the sheaf of isomorphism classes of integrable connections of rank $r$. We then define the moduli space of integrable connections on $X$ and the moduli space of $v$-vector bundles on $X$ with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α$, and prove that they are small $v$-stacks in the sense of Scholze. These constructions generalize Heuer's work on $p$-adic Simpson correspondence. |
| title | On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2403.01363 |