Moduli spaces in $p$-adic non-abelian Hodge theory
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917583600484352 |
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| author | Heuer, Ben |
| author_facet | Heuer, Ben |
| contents | We propose a new moduli-theoretic approach to the $p$-adic Simpson correspondence for a smooth proper rigid space $X$ over $\mathbb C_p$ with coefficients in any rigid analytic group $G$, in terms of a comparison of moduli stacks. For its formulation, we introduce the class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative $p$-adic Hodge theory. For any smoothoid space $Y$, we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphism \[R^1ν_{\ast}G\xrightarrow{\sim} \mathrm{Higgs}_G\] where $ν:Y_{v}\to Y_{et}$ is the natural morphism of sites, and where $\mathrm{Higgs}_G$ is the sheaf of isomorphism classes of $G$-Higgs bundles on $Y_{et}$. We also prove a generalisation of Faltings' local $p$-adic Simpson correspondence to $G$-bundles and to perfectoid families.
We apply these results to deduce $v$-descent criteria for étale $G$-bundles which show that $G$-Higgs bundles on $X$ form a small $v$-stack $\mathscr Higgs_G$. As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism $\mathscr Bun_{G,v}\to \mathcal A_G$ from the small $v$-stack of $v$-topological $G$-bundles on $X$ to the Hitchin base. This allows us to give a conjectural reformulation of the $p$-adic Simpson correspondence for $X$ in a more geometric and more canonical way, namely in terms of a comparison of Hitchin morphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13819 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Moduli spaces in $p$-adic non-abelian Hodge theory Heuer, Ben Algebraic Geometry 14D22, 14G45, 14G22, 14D10 We propose a new moduli-theoretic approach to the $p$-adic Simpson correspondence for a smooth proper rigid space $X$ over $\mathbb C_p$ with coefficients in any rigid analytic group $G$, in terms of a comparison of moduli stacks. For its formulation, we introduce the class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative $p$-adic Hodge theory. For any smoothoid space $Y$, we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphism \[R^1ν_{\ast}G\xrightarrow{\sim} \mathrm{Higgs}_G\] where $ν:Y_{v}\to Y_{et}$ is the natural morphism of sites, and where $\mathrm{Higgs}_G$ is the sheaf of isomorphism classes of $G$-Higgs bundles on $Y_{et}$. We also prove a generalisation of Faltings' local $p$-adic Simpson correspondence to $G$-bundles and to perfectoid families. We apply these results to deduce $v$-descent criteria for étale $G$-bundles which show that $G$-Higgs bundles on $X$ form a small $v$-stack $\mathscr Higgs_G$. As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism $\mathscr Bun_{G,v}\to \mathcal A_G$ from the small $v$-stack of $v$-topological $G$-bundles on $X$ to the Hitchin base. This allows us to give a conjectural reformulation of the $p$-adic Simpson correspondence for $X$ in a more geometric and more canonical way, namely in terms of a comparison of Hitchin morphisms. |
| title | Moduli spaces in $p$-adic non-abelian Hodge theory |
| topic | Algebraic Geometry 14D22, 14G45, 14G22, 14D10 |
| url | https://arxiv.org/abs/2207.13819 |