Rigid Analytic Varieties Research Articles

On 𝑝-adic Simpson and Riemann–Hilbert correspondences in the imperfect residue field case

Let K K be a mixed characteristic complete discrete valuation field with residue field admitting a finite p p -basis, and let G K G_K be the Galois group. Inspired by Liu and Zhu’s construction of p p -adic Simpson and Riemann–Hilbert correspondences over rigid analytic varieties, we construct such correspondences for representations of G K G_K . As an application, we prove a Hodge–Tate (resp. de Rham) “rigidity” theorem for p p -adic representations of G K G_K , generalizing a result of Morita.

Étale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases

We prove a finiteness theorem and a comparison theorem in the theory of étale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension ≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le 1$$\\end{document}, the compactly supported higher direct image preserves quasi-constructibility. Though the analogous statement for morphisms with higher dimensional target fails in general, we prove that, in the algebraizable case, it holds after replacing the target with a modification. We deduce it from a known finiteness result in the theory of nearby cycles over general bases and a new comparison result, which gives an identification of the compactly supported higher direct image sheaves, up to modification of the target, in terms of nearby cycles over general bases.

A note on Hodge–Tate spectral sequences

AbstractWe prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$ -cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$ -cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.

Faithfully Flat Descent of Quasi-Coherent Complexes on Rigid Analytic Varieties via Condensed Mathematics

Abstract Faithfully flat descent of pseudo-coherent complexes in rigid geometry was proved by Mathew. In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic varieties, which have been introduced by Clausen and Scholze by means of condensed mathematics.

On affinoids in quotients of Fermat varieties and explicit formula for Jacobi sum Hecke characters

By regarding the Fermat variety over a local field as a rigid analytic variety, we construct a family of affinoids in quotients of the Fermat varieties and compute the reduction of them. As a result, we explicitly compute the middle cohomology of the Fermat variety as a Galois representation. As a byproduct, we give an explicit formula for the ramified components of Jacobi sum Hecke characters in many variables.

UNIFORM LOCAL CONSTANCY OF ÉTALE COHOMOLOGY OF RIGID ANALYTIC VARIETIES

AbstractWe prove some $\ell $ -independence results on local constancy of étale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighbourhood in the analytic topology such that, for every prime number $\ell $ different from the residue characteristic, the closed subscheme and the open neighbourhood have the same étale cohomology with ${\mathbb Z}/\ell {\mathbb Z}$ -coefficients. The existence of such an open neighbourhood for each $\ell $ was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.

De Rham Comparison and Poincaré Duality for Rigid Varieties

Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham \({\mathbb {Z}}_p\)-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any \({\mathbb {Z}}_p\)-local systems, and establish the Poincaré duality for such cohomology after inverting p.

A -adic monodromy theorem for de Rham local systems

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a$p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.

Vector Bundles with Numerically Flat Reduction on Rigid Analytic Varieties andp-Adic Local Systems

AbstractWe show how to functorially attach continuous $p$-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over ${{\mathbb {C}}}_p$. This generalizes results by Deninger and Werner for vector bundles on smooth algebraic varieties. Our approach uses fundamental results on the pro-étale site of a rigid analytic variety introduced by Scholze. This enables us to get rid of the smoothness condition and to work in the analytic category. Moreover, under some mild conditions, the functor we construct gives a full embedding of the category of vector bundles with numerically flat reduction into the category of continuous ${{\mathbb {C}}}_p$-representations. This provides new insights into the $p$-adic Simpson correspondence in the case of a vanishing Higgs field.

A counterexample to an optimistic guess about étale local systems

In p-adic Hodge theory, it is known that if a Galois representation is de Rham, then it becomes semistable after extension of the base field. Liu and Zhu asked whether a corresponding result holds in the relative setting: given an \'etale local system on a quasi-compact rigid analytic variety (for example, a projective scheme) over a p-adic field, does it become semistable after a finite extension of the base field? We give a counterexample.

Non-archimedean hyperbolicity and applications

Abstract Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K-analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.

Extending meromorphic connections to coadmissible ̑𝒟-modules

Abstract We investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D ⏜ X \overparen{\mathcal{D}}_{X} -module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.

Rigid-analytic varieties with projective reduction violating Hodge symmetry

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li.

A homotopy exact sequence for overconvergent isocrystals

AbstractIn this article we prove exactness of the homotopy sequence of overconvergent fundamental groups for a smooth and projective morphism in characteristicp. We do so by first proving a corresponding result for rigid analytic varieties in characteristic$0$, following dos Santos [dS15] in the algebraic case. In characteristicp, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result. We then use this to deduce a Lefschetz hyperplane theorem for convergent fundamental groups, as well as a comparison theorem with the étale fundamental group.

Line bundles on rigid varieties and Hodge symmetry

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space.

Vanishing and comparison theorems in rigid analytic geometry

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

RIGIDITY FOR RIGID ANALYTIC MOTIVES

AbstractIn this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field $K$. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to $\mathbb{Z}[1/p]$-coefficients.

The Berkovich realization for rigid analytic motives

We prove that the functor associating to a rigid analytic variety the singular complex of the underlying Berkovich topological space is motivic, and defines the maximal Artin quotient of a motive. We use this to generalize Berkovich's results on the weight-zero part of the étale cohomology of a variety defined over a non-archimedean valued field.

Virtual rigid motives of semi-algebraic sets

Let k be a field of characteristic zero containing all roots of unity and $$K=k(( t))$$ . We build a ring morphism from the Grothendieck ring of semi-algebraic sets over K to the Grothendieck ring of motives of rigid analytic varieties over K. It extends the morphism sending the class of an algebraic variety over K to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan’s motivic integration and Ayoub’s equivalence between motives of rigid analytic varieties over K and quasi-unipotent motives over k; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.

LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

In this note, we prove the logarithmic$p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally$K(\unicode[STIX]{x1D70B},1)$(in a certain sense) with respect to$\mathbb{F}_{p}$-local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.