Line bundles on rigid spaces in the $v$-topology

Heuer, Ben

doi:10.48550/arxiv.2012.07918

Cited by 7 publications

(19 citation statements)

References 8 publications

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“…One case in which a p-adic Corlette-Simpson correspondence for representations is known is the case of rank one [20]: In this case, characters of the étale fundamental group correspond to pro-finite-étale Higgs line bundles, and one can also explicitly describe pro-finite-étale line bundles as topological torsion points in the Picard variety [18]. However, a "full" p-adic Corlette-Simpson correspondence from a specific subcategory of Higgs bundles on a proper smooth variety X to the category of finite-dimensional continuous K -linear representations of the étale fundamental group of X has so far not been established yet, not even in non-trivial special cases of X .…”

Section: Naturality Of the Correspondencementioning

confidence: 99%

“…It is easy to see that X is the universal pro-finite-étale cover of X in the following sense: [20,Lemma 4.8] Let Y → X be any pro-finite-étale cover with a lift y ∈ Y (K ) of x. Then there is a unique morphism X → Y over X sending x to y.…”

Section: The Diamantine Universal Covermentioning

confidence: 99%

“…From H 0 ( X , O) = K it is clear that the category of finite free O-modules on X is equivalent to the category of finite K -vector spaces. Since X → X is a π 1 (X , x)-torsor, it follows that giving a descent datum of a finite free O-module M along X → X amounts to specifying a π 1 (X , x)-action on M, this can be seen as an instance of the Cartan-Leray spectral sequence [20,Proposition 3.6].…”

Section: Lemma 28mentioning

confidence: 99%

“…We may thus assume that ρ : T p A → K × is a character. In this case, the Corollary follows from the results of[20]: As recalled in Sect. 2.6, the v-line bundle V ρ is analytic if and only if HT log(V ρ ) = 0.…”

mentioning

confidence: 93%

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The p-adic Corlette–Simpson correspondence for abeloids

2022

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For an abeloid variety A over a complete algebraically closed field extension K of $$\mathbb {Q}_p$$ Q p , we construct a p-adic Corlette–Simpson correspondence, namely an equivalence between finite-dimensional continuous K-linear representations of the Tate module and a certain subcategory of the Higgs bundles on A. To do so, our central object of study is the category of vector bundles for the v-topology on the diamond associated to A. We prove that any pro-finite-étale v-vector bundle can be built from pro-finite-étale v-line bundles and unipotent v-bundles. To describe the latter, we extend the theory of universal vector extensions to the v-topology and use this to generalise a result of Brion by relating unipotent v-bundles on abeloids to representations of vector groups.

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Section: Naturality Of the Correspondencementioning

confidence: 99%

Section: The Diamantine Universal Covermentioning

confidence: 99%

Section: Lemma 28mentioning

confidence: 99%

mentioning

confidence: 93%

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The p-adic Corlette–Simpson correspondence for abeloids

2022

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“…Line bundles. This was essentially done in previous work of the first author [Heu21c] (see Section 2.6).…”

Section: Introductionmentioning

confidence: 99%

The $p$-adic Corlette-Simpson correspondence for abeloids

Heuer,

Mann,

Werner

2021

Preprint

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For an abeloid variety A over a complete algebraically closed field extension K of Qp, we construct a p-adic Corlette-Simpson correspondence, namely an equivalence between finite-dimensional continuous K-linear representations of the Tate module and a certain subcategory of the Higgs bundles on A. To do so, our central object of study is the category of vector bundles for the v-topology on the diamond associated to A. We prove that any pro-finite-étale v-vector bundle can be built from pro-finite-étale v-line bundles and unipotent v-bundles. To describe the latter, we extend the theory of universal vector extensions to the v-topology and use this to generalize a result of Brion by relating unipotent v-bundles on abeloids to representations of vector groups.

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Diamantine Picard functors of rigid spaces

Heuer¹

2021

Preprint

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For a connected smooth proper rigid space X over a perfectoid field extension of Qp, we show that the Picard functor of X ♦ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid group variety if and only if the rigid analytic Picard functor is.As an application, we determine which line bundles are trivialized by pro-finiteétale covers, and prove unconditionally that the associated "topological torsion Picard functor" is represented by a divisible analytic group. We use this to generalize and geometrize a construction of Deninger-Werner in the p-adic Simpson correspondence: There is an isomorphism of rigid analytic group varieties between the moduli space of continuous characters of π1(X, x) and that of pro-finite-étale Higgs line bundles on X.This article is part II of a series about line bundles on rigid spaces as diamonds.1. The natural map (Pic X,ét ) ♦ → Pic ♦ X,ét is an isomorphism of sheaves on Perf K,ét .2. If moreover K is algebraically closed, then the v-Picard functor Pic ♦ X,v fits into a split exact sequence of abelian sheaves on Perf K,ét 0 → Pic ♦ X,ét → Pic ♦ X,v → H 0 (X, Ω 1 )(−1) ⊗ K G a → 0.

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Line bundles on rigid spaces in the $v$-topology

Cited by 7 publications

References 8 publications

The p-adic Corlette–Simpson correspondence for abeloids

The p-adic Corlette–Simpson correspondence for abeloids

The $p$-adic Corlette-Simpson correspondence for abeloids

Diamantine Picard functors of rigid spaces

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