272 pages. This is the final version, to appear as a memoir in the Astérisque collection. I am again very grateful to the referee for his/her incredibly careful reading of the revised version
This text is devoted to the systematic study of relative properties in the context of Berkovich analytic spaces. We first develop a theory of flatness in this setting. After having shown through a counter-example that naive flatness cannot be the right notion because it is not stable under base change, we define flatness by {\em requiring} invariance under base change, and we study a first important class of flat morphisms, that of quasi-smooth ones. We then show the existence of local {\em dévissages} (in the spirit of Raynaud and Gruson) for coherent sheaves, which we use, together with a study of the local rings of "generic fibers" of morphisms, to prove that a {\em boundaryless}, naively flat morphism is flat. After that we prove that the image of a compact analytic space by a flat morphism can be covered by a compact, relatively Cohen-Macaulay and zero-dimensional multisection, and the image of the latter is shown to be a compact analytic domain of the target; we thus recover the result by Raynaud telling that the image of a compact analytic space under a flat morphism is a compact analytic domain of the target. In the last part of this work we study some validity loci. We first prove that the flatness locus of a given morphism of analytic spaces is a Zariski-open subset of the source. We then look at the {\em fiberwise} validity loci of the usual commutative algebra properties. We prove that the results we could expect actually hold: the fiberwise validity locus of such a property is Zariski-constructible, and Zariski-open under suitable extra assumptions (flatness, and also sometimes equidimentionality).
A Poincaré lemma for real-valued differential forms on Berkovich spaces
Berkovich解析空间上的实值微分形式由Chambert-Loir和Ducros在“Formes différentielles réelles et courants sur les espaces de Berkovich”中使用多面体复形上的超形引入。我们证明了一个庞加莱引理，这些superfect和使用它也证明了庞加莱引理的实值微分形式的Berkovich空间。对于superforms，我们进一步显示有限维的相关的德拉姆上同调多面体复杂的所有（双）度。我们还显示了有限维的实值德拉姆上同调的分析代数簇在一些bidegrees。
We prove some base change theorems for coherent cohomology in the setting of Berkovich spaces. In this setting, we get a flat base change theorem, and some proper base change theorems that are analogue to theorems from scheme theory.
We develop in this article flattening techniques for coherent sheaves in the realm of Berkovich spaces; we are inspired by the general strategy that Raynaud and Gruson have used for dealing with the analogous problem in scheme theory. As an application, we give a general description of the image of an arbitrary morphism between compact analytic spaces.
Utilisation de l'aplatissement en géométrie de Berkovich
Sorbonne Université（索邦大学）；Université Paris-Diderot（巴黎第七大学）；CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG（法国国家科学研究中心，儒勒·亨利·庞加莱数学研究所-巴黎左岸）
Let $K$ be a non-archimedean local field, $X$ a smooth and proper $K$-scheme, and fix a pluricanonical form on $X$. For every finite extension $K'$ of $K$, the pluricanonical form induces a measure on the $K'$-analytic manifold $X(K')$. We prove that, when $K'$ runs through all finite tame extensions of $K$, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of $X$ converge to a Lebesgue-type measure on the temperate part of the Kontsevich--Soibelman skeleton, assuming the existence of a strict normal crossings model for $X$. We also prove a similar …
This article studies descent theory in the setting of Berkovich spaces. We give sufficient conditions for a given fibered category over the category of k-affinoid algebras to be a stack for the Berkovich analogue of the faithfully-flat topology. We give some applications to the faithfully flat descent of morphisms and show that some descent data are always effective. We also show that the property of being algebraic for a morphism between the analytification of two schemes is a local property for the faithfully-flat topology.
In non-archimedean setting, we establish a Lehto--Virtanen-type theorem for a morphism from the punctured Berkovich closed unit disk $\overline{\mathsf{D}}\setminus\{0\}$ in the Berkovich affine line to the Berkovich projective line $\mathsf{P}^1$ having an isolated essential singularity at the origin, and then establish a big Picard-type theorem for such an open subset $Ω$ in the Berkovich projective space $\mathsf{P}^N$ of any dimension $N$ that the family of all morphisms from $\overline{\mathsf{D}}\setminus\{0\}$ to $Ω$ is normal in a non-archimedean Montel's sense. As an application of th…
We consider an algebraic variety X together with the choice of a subvariety Z. We show that any coherent sheaf on X can be constructed out of a coherent sheaf on the formal neighborhood of Z, a coherent sheaf on the complement of Z, and an isomorphism between certain representative images of these two sheaves in the category of coherent sheaves on a Berkovich analytic space W which we define.
Les squelettes accessibles d'un espace de Berkovich
Sorbonne Université（索邦大学）；Université Paris-Diderot（巴黎第七大学）；CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG（法国国家科学研究中心，儒勒·庞加莱数学研究所-巴黎左岸，IMJ-PRG）；Institut Camille Jordan, Université de Lyon 1（卡米尔·若当研究所，里昂第一大学）
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.
We develop properties of unramified, étale and smooth morphisms between Berkovich spaces over $\mathbb{Z}$. We prove that they satisfy properties analogous to those of morphisms of schemes and we provide analytification criteria. Our results hold for any valued field, rings of integers of a number field and discrete valuation rings. Those cases are treated by a unified way.
We study tropical Dolbeault cohomology for Berkovich analytic spaces, as defined by Chambert-Loir and Ducros. We provide a construction that lets us pull back classes in tropical cohomology to classes in tropical Dolbeault cohomology as well as check whether those classes are non-trivial. We further define tropical cohomology with integral coefficients on the Berkovich space and provide some computations. Our main tool is extended tropicalization of toric varieties as introduced by Kajiwara and Payne.
This text is an exposition of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It consists of two parts, the first one of an introductory nature, and the second one more advanced. The first part is meant to be an introduction to the theory of Berkovich spaces focused on the case of the affine line. We define the Berkovich affine line and present its main properties, with many details: classification of points, path-connectedness, metric structure, variation of rational functions, etc. Contrary to many other introductory texts, we do not assume th…
We prove a folklore conjecture that the Bergman measure along a holomorphic family of curves parametrized by the punctured unit disk converges to the Zhang measure on the associated Berkovich space. The convergence takes place on a Berkovich hybrid space. We also study the convergence of the Bergman measure to a measure on a metrized curve complex in the sense of Amini and Baker.
In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by André, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language o…
Let $k$ be a non-archimedean complete field. We prove a substitute for the reduced fiber theorem (of Bosch, Lütkebohmert and Raynaud) that holds for every morphism $Y\to X$ flat and with geometrically reduced fibers between $k$-affinoid spaces in the sense of Berkovich, without assuming that $X$ and $Y$ are strict, nor that the relative dimension of $Y$ over $X$ is constant. We do not use the original reduced fiber theorem, nor the language or the techniques of formal geometry. Our statement is formulated in terms of Temkin's graded reduction; our proof rests on a finiteness result of Grauert …
In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the second type, including the descent of properties of being a good analytic space and being a morphism without boundary.
We study the tropicalization of the moduli space of algebraic spin curves, exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves, prove that it is isomorphic to the skeleton of the Berkovich analytification of the moduli space of algebraic spin curves, and give a geometric interpretation of the retraction map in terms of a tropicalization map.
