The p-adic Simpson Correspondence (AM-193)

Abbes, Ahmed; Gros, M.; Tsuji, Takeshi

doi:10.23943/princeton/9780691170282.001.0001

Cited by 31 publications

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“…Dans la suite de cette section, on se donne une S-courbe semi-stable et régulière X que l'on munit de la structure logarithmique M X définie par sa fibre spéciale X s . Le morphisme de schémas logarithmiques f : (X, M X ) → (S, M S ) est alors adéquat au sens de [4,III.4.7]. On notera que X η est le sous-schéma ouvert maximal de X où la structure logarithmique M X est triviale.…”

Section: Correspondance De Simpsonunclassified

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Transport Parallèle Et Correspondance De Simpson -Adique

2017

Forum of Mathematics, Sigma

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RésuméDeninger et Werner ont développé un analogue pour les courbes p-adiques de la correspondance classique de Narasimhan et Seshadri entre les fibrés vectoriels stables de degré 0 et les représentations unitaires du groupe fondamental topologique pour une courbe complexe propre et lisse. Par transport parallèle, ils ont associé fonctoriellementà chaque fibré vectoriel sur une courbe p-adique, dont la réduction est fortement semi-stable de degré 0, une représentation p-adique du groupe fondamental de la courbe. Ils se sont posé quelques questions : leur foncteur est-il pleinement fidèle ? La cohomologie des systèmes locaux fournis par celui-ci admet-elle une filtration de Hodge-Tate ? Leur construction est-elle compatible avec la correspondance de Simpson p-adique développée par Faltings ? Nous répondonsà ces questions dans cet article. AbstractDeninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree 0 and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this functor admits a Hodge-Tate filtration; and whether their construction is compatible with the p-adic Simpson correspondence developed by Faltings. We answer these questions in this article.

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Section: Correspondance De Simpsonunclassified

“…. , n} de N. On munit C /U × [n] de la topologie totale relative au site fibré constant [4,III.7.9], on a uneéquivalence canonique de topos On en déduit un isomorphisme canonique fonctoriel …”

Section: Mod(a)unclassified

Transport Parallèle Et Correspondance De Simpson -Adique

2017

Forum of Mathematics, Sigma

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“…(i) Faltings site. Faltings’ construction of the period morphism uses an auxiliary topos, a topos of ‘sheaves of local systems’ (see [Fal89, III] and [Fal02, 3]), that is now known as the ‘Faltings topos’ (a term coined by Abbes and Gros [AG16]). We will briefly describe it.…”

Section: Comparison Of Period Morphismsmentioning

confidence: 99%

On uniqueness ofp-adic period morphisms, II

Nizioł¹

2020

Compositio Math.

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We prove equality of the various rational $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterion we had found earlier for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined (and up to a change of Hyodo–Kato cohomology). We do it by lifting the syntomic and almost étale period morphisms to the $h$-site of varieties over a field, where their equality with the $h$-cohomology period morphism can be checked directly using the Beilinson Poincaré lemma and the case of dimension $0$. This also shows that the syntomic and almost étale period morphisms have a natural extension to the Voevodsky triangulated category of motives and enjoy many useful properties (since so does the $h$-cohomology period morphism).

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“…There exists also a p-adic version of Simpson's correspondence developed by Faltings [Fal05] and [Fal11]. A detailed and systematic treatment is provided by [AGT16]. If X is a curve, this correspondence relates Higgs bundles and generalized representations.…”

Section: Introductionmentioning

confidence: 99%

Parallel transport for vector bundles on 𝑝-adic varieties

Deninger¹,

Werner

2019

J. Algebraic Geom.

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We develop a theory ofétale parallel transport for vector bundles with numerically flat reduction on a p-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p-adic representation of theétale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings' p-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations. MSC: 14J20, 11G252 Numerically flat bundles Throughout this paper, a variety over a field k is a geometrically integral separated scheme of finite type over k. Recall that a scheme is called integral if it is irreducible and reduced.We want to study the following class of vector bundles which was introduced in [DMOS82] (2.34).Definition 2.1 Let X be a connected and complete scheme over a perfect field k. A vector bundle E over X is called Nori-semistable if for all k-morphisms f : C → X from smooth connected projective curves C over k into X the pullback f * E is semistable of degree zero.A related but different notion is due to Nori [Nor82] who considers only pullbacks to embedded smooth projective curves. In [DMOS82] the bundles in Definition 1 are simply called semistable. Since this can lead to misunderstandings, the name Nori-semistable has been adopted by several authors.

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The p-adic Simpson Correspondence (AM-193)

Cited by 31 publications

References 0 publications

Transport Parallèle Et Correspondance De Simpson -Adique

Transport Parallèle Et Correspondance De Simpson -Adique

On uniqueness ofp-adic period morphisms, II

Parallel transport for vector bundles on 𝑝-adic varieties

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