Formal and rigid geometry: an intuitive introduction, and some applications

通过形式模型的刚性解析空间的紧化

Compactifications of rigid analytic spaces through formal models

本文利用形式模型与相对Riemann-Zariski空间，为刚性解析空间的结构态射构造了Huber万有紧化的新方法。

特殊刚性簇上的动机积分与动机积分恒等式猜想

Motivic integration on special rigid varieties and the motivic integral identity conjecture

本文通过发展特殊刚性簇上的等变动机积分新框架，证明了Kontsevich和Soibelman的动机积分恒等式猜想，并建立了具有Fubini型性质的动机体积概念。

Affine manifolds are rigid analytic spaces in characteristic one, I

I extend the definitions of schemes relative to monoids with zero - and therefore, toric geometry - to the world of formal schemes. This expands the usual framework to include, for instance, models for Mumford's degenerating Abelian varieties. Following the usual toric paradigm, normal formal monoid schemes can be classified in terms of certain cone complexes, and their properties understood in combinatorial terms. I use this to give a simple algebraisation criterion. I also reformulate the traditional notions of separated and proper morphism in a manner amenable to the context of relative for…

A $p$-Adic 6-Functor Formalism in Rigid-Analytic Geometry

We develop a full 6-functor formalism for $p$-torsion étale sheaves in rigid-analytic geometry. More concretely, we use the recently developed condensed mathematics by Clausen--Scholze to associate to every small v-stack (e.g. rigid-analytic variety) $X$ with pseudouniformizer $π$ an $\infty$-category $\mathcal D^a_\square(\mathcal O^+_X/π)$ of "derived quasicoherent complete topological $\mathcal O^+_X/π$-modules" on $X$. We then construct the six functors $\otimes$, $\underline{Hom}$, $f^*$, $f_*$, $f_!$ and $f^!$ in this setting and show that they satisfy all the expected compatibilities, s…

Motivic cohomology of fat points in Milnor range via formal and rigid geometries

We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings $k[t]/(t^m)$ with $m \geq 2$, in one variable over a field $k$. We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor $K$-theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor $K$-groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.

Foundations of Rigid Geometry I

In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian (cf. introduction). The manuscript is encyclopedic and almost self-contained, and contains plenty of new results. A discussion on relationship with J. Tate's rigid analytic geometry, V. Berkovich's analytic geometry and R. H…

On the formal principle for curves on projective surfaces

We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.

Deformation theory of rigid-analytic spaces

In this paper, we study a deformation theory of rigid analytic spaces. We develop a theory of cotangent complexes for rigid geometry which fits in with our deformations. We then use the complexes to give a cohomological description of infinitesimal deformation of rigid analytic spaces. Moreover, we will prove an existence of a formal versal family for a proper rigid analytic space.

Derived $\mathcal{O}_k$-adic geometry and derived Raynaud localization theorem

The goal of the present text is to state and prove a generalization of Raynaud localization theorem in the setting of derived geometry. More explicitly, we show that the $\infty$-category of quasi-paracompact and quasi-separated derived $k$-analytic spaces can be realized as a localization of the $\infty$-category of admissible derived formal schemes. We construct a derived rigidification functor generalizing Raynaud rigidification functor. In order to construct the latter we will need to formalize derived formal $\mathcal{O}_k$-adic formal geometry via a structured spaces approach. We prove t…

Altered Local Uniformization Of Rigid-Analytic Spaces

We prove a version of Temkin's local altered uniformization theorem. We show that for any rig-smooth, quasi-compact and quasi-separated admissible formal $\mathcal{O}_K$-model $\mathfrak{X}$, there is a finite extension $K'/K$ such that $\mathfrak{X}_{\mathcal{O}_{K'}}$ locally admits a rig-étale morphism $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and a rig-isomorphism $h\colon \mathfrak{X}" \to \mathfrak{X}'$ with $\mathfrak{X}'$ being a successive semi-stable curve fibration over $\mathcal{O}_{K'}$ and $\mathfrak{X}"$ being a poly-stable formal $\mathcal{O}_{K'}$-scheme. Mor…

$p$-adic权重谱序列的算术$\mathcal{D}$-模理论研究：严格半稳定方案在形式幂级数环上的构造与性质

$p$-Adic Weight Spectral Sequences of Strictly Semi-stable Schemes over Formal Power Series Rings via Arithmetic $\mathcal{D}$-modules

本文通过算术$\mathcal{D}$-模理论，构建了严格半稳定方案在形式幂级数环上的$p$-adic权重谱序列，其$E_1$项由闭纤维的不可约成分的刚性上同调描述，$E_\infty$项猜想由Lazda-Pál的刚性上同调的unipotent邻近循环描述，并探讨了其泛函性。

Rigid cohomology of locally noetherian schemes Part 2 : Crystals

We introduce the general notions of an overconvergent site and a constructible crystal on an overconvergent site. We show that if $V$ is a geometric materialization of a locally noetherian formal scheme $X$ over an analytic space $O$ defined over $\mathbb Q$, then the category of constructible crystals on $X/O$ is equivalent to the category of constructible modules endowed with an overconvergent connection on the tube $\,]X[_V$ of $X$ in $V$. We also show that the cohomology of a constructible crystal is then isomorphic to the de Rham cohomology of its realization on the tube $\,]X[_V$. This i…

The six functors for Zariski-constructible sheaves in rigid geometry

We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well-adapted to "spreading out" arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.

刚性解析1-动机与阿贝尔簇的共轭单值化

Rigid analytic 1-motives and conjugate uniformization of abeloid varieties

本文针对p进域上的阿贝尔簇，研究刚性解析1-动机理论及阿贝尔簇的共轭单值化，建立了形式与刚性解析1-动机的等价性等结果，还构造了相关p可除群。

On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields

We investigate the action of the Weil group on the compactly supported l-adic cohomology groups of rigid spaces over local fields. We prove that every eigenvalue of the action is a Weil number when either a rigid space is smooth or the characteristic of the base field is equal to 0. Since a smooth rigid space is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme, we can use the techniques of alterations and monodromy spectral sequences in the smooth case. In the general case, we use the continuity theorem of Huber, which requires the restriction on the characteri…

刚性解析周期映射、Lubin-Tate 空间与稳定同伦论

The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory

本文利用从形变空间到射影空间的 étale 刚性解析映射研究 Lubin-Tate 空间几何，得到等变典范丛的简单描述，并由此导出稳定同伦论中对偶化复形的公式。

刚性几何中的解析 de Rham 栈

The analytic de Rham stack in rigid geometry

本文利用Clausen与Scholze的解析栈理论引入导出Tate adic空间，定义刚性几何中的解析de Rham栈，将D-cap-模推广为解析D-模，并建立六函子形式体系与Poincaré对偶，关联p-adic李群的固态局部解析表示。

几乎相干模与几乎相干sheaves

Almost coherent modules and almost coherent sheaves

本文介绍了几乎相干模和sheaves理论，展示了其与常规相干sheaves相似的性质，并构造了Grothendieck扭曲图像函子的几乎版本，证明了其性质。

在六 functor 形式主义中研究光滑范畴和解析几何中核范畴的紧生成

Smooth categories in a 6 functor formalism and compact generation for nuclear categories in analytic geometry

本文在六 functor 形式主义框架下研究光滑 ∞-范畴，并利用凝聚数学理论和解析堆的结果，证明了刚性解析流形光滑当且仅当其关联的核sheaves范畴光滑，同时将核sheaves范畴的紧生成性与刚性解析流形的代数化联系起来，从而得到一个非原子生成但内部光滑的范畴例子。

相对 adic 空间的形式模型

Formal models for relative adic spaces

本文将 Raynaud 的形式模型理论从刚性解析空间扩展到任意 Tate 仿元基上的一致 qcqs adic 空间，并引入规范形式爆破作为关键新概念。

Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve

We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.

Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence

This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety $X$ is formally integrable at any point, and, if we suppose that its singular locus has codimension $\geq 2$, then the truncation of its analytification is a locally inte…

刚性解析簇的有理$p$-进霍奇理论

Rational $p$-adic Hodge theory for rigid-analytic varieties

本文建立$\boldsymbol{\rm C}_p$上无紧性/光滑性假设的刚性解析簇有理$p$-进霍奇理论，通过凝聚与固体形式体系证明其与多类$p$-进上同调的比较关系，验证Le Bras猜想并推广Colmez-Niziol的结果。