Moduli spaces in $p$-adic non-abelian Hodge theory

Abstract

We propose a new moduli theoretic approach to the p-adic Simpson correspondence for a smooth proper rigid space X with coefficients in any rigid analytic group G, in terms of a comparison of moduli stacks. For its formulation, we introduce a 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 isomorphismwhere ν : Yv → Y ét is the natural morphism of sites, and where Higgs G is the sheaf of isomorphism classes of G-Higgs bundles on Y ét . 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 H iggs G . As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism BunG,v → AG from the small v-stack of v-topological G-bundles on X to the Hitchin base. This allows us to reformulate the conjectural p-adic Simpson correspondence for X in a more geometric and more canonical way as a comparison of Hitchin morphisms.

The p-adic Corlette–Simpson correspondence for abeloids

“…There is for any smooth rigid space U an intrinsic notion of "smallness" for both pro-étale vector bundles and Higgs bundles. As we will not need the technical details, we just refer to [Heu22d,§6] for the definition. What will be important for us is only the following:…”

“…Let (E ′ , θ ′ ) = LS f ′ (V ), then there exists a non-canonical isomorphism between (E, θ) and (E ′ , θ ′ ) after étale localisation on U , e.g. by [Heu22d,Thm 1.2]. Indeed, we can see this via twisting: Let B be the coherent quotient of (θ, θ ′ , 0) : T X → End((E, θ) ⊕ (E ′ , θ ′ ) ⊕ ( Ω, 0)) from Definition 4.1 and let B = ν * B.…”

“…For this, we combine ideas of [AHLB23] and [Heu22d]: Let J := ker(T X θ − → End(E)) and consider T J := lim ← −n T X /J n . Since the natural map T X → T J is flat, we have Ext TX (O X , E) = Ext TJ (O X , E).…”

“…At this point, it suffices to show that every class in Ext n X proét (O X , V ) is of the form (13). To see this, we use that by [Heu22d, with O( U ). Here the action of γ i ∈ ∆ on the left is given by multiplication with T i + 1 (see for example the proof of [Sch13a]).…”

“…Strategy. The approach of this article is rooted in the idea of using p-adic analytic moduli spaces to study the p-adic Simpson correspondence, initiated in [Heu22b] [Heu22d].…”