论文标题
$ p $ - 阿迪亚非亚伯式霍奇理论中的模量空间
Moduli spaces in $p$-adic non-abelian Hodge theory
论文作者
论文摘要
我们为$ \ \ Mathbb c_p $ a $ p $ adic simpson的通信提出了一种新的模量理论方法。为了表达，我们介绍了一系列“平滑场空间”，它们是光滑的刚性空间的完美体族，非常适合研究相对$ p $ $ addic hodge理论。对于任何平滑的空间$ y $，然后我们构建一个“隔离的非亚伯利亚杂货通讯”，即规范同构\ [r^1ν_ {\ ast} g \ xrightArrow {\ sim} \ sim} \ sim} \ simrm {higgs}} $}网站的形态，以及$ \ mathrm {higgs} _g $的位置是$ g $ -higgs捆绑$ y_ {et} $的同构类别的捆绑。我们还证明了Faltings的本地$ P $ -ADIC SIMPSON对$ G $捆绑以及与Perfectoid家庭的概括。 我们将这些结果应用于推断$ v $ deScent的标准，用于étale$ g $ - 捆绑包，这些标准表明$ g $ -higgs捆绑在$ x $ a $ x $ forem a a小$ v $ -stack $ \ mathscr higgs_g $。作为第二个应用程序，我们在贝蒂方面构建了Hitchin Morphism的类似物：一种形态$ \ Mathscr bun_ {g，v} \ to \ Mathcal a_g $ to $ v $ v $ v $ v $ - 多$ v $ spack of $ v $ toprogical $ g $ g $ g $ g $ x $ to $ x $ to the Hitchin base上的$ x $。这使我们能够以$ x $的$ p $ adic simpson对$ x $的猜想进行重新进行，以一种更加几何和更具典型的方式，即在比较希钦形态的比较方面。
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.
