non-Archimedean Analytic Geometry Research Articles

Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields. The topics of the talks included foundational results on analytic spaces as well as applications to the local Langlands conjecture, birational geometry, p-adic cohomology theories, Shimura varieties and the non-Archimedean Simpson correspondence.

An Intrinsic Characterization of Bruhat–Tits Buildings Inside Analytic Groups

Given a reductive group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat–Tits building into the analytic space associated with the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

Derived Hom Spaces in Rigid Analytic Geometry

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic information of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us with sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also merit independent interest. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov–Witten invariants.

Representability theorem in derived analytic geometry

We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived non-archimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor. So the theorem confirms that the notion of derived analytic space is natural and sufficiently general. On the other hand, the conditions are easy to verify in practice. So the theorem enables us to enhance various classical moduli spaces with derived structures, thus provides plenty of down-to-earth examples of derived analytic spaces. For the purpose of proof, we study analytification, square-zero extensions, analytic modules and cotangent complexes in the context of derived analytic geometry. We will explore applications of the representability theorem in our subsequent works. In particular, we will establish the existence of derived mapping stacks via the representability theorem.

Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

Canonical measures on metric graphs and a Kazhdan’s theorem

We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a "trace formula". In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.

Wild coverings of Berkovich curves

This paper is an extended version of the author’s talk given at the conference ”Non-Archimedean analytic geometry: theory and practice” held in August 2015 at Papeete, and I wish to thank the organizers. It gives a brief overview of results and methods of works [CTT14] and [Tem14] on the structure of finite morphisms between Berkovich curves. The structure of tame morphisms between smooth Berkovich curves is pretty well known and it is completely controlled by the simultaneous semistable reduction theorem, see, for example, [ABBR13]. The structure of wild morphisms was for a long time terra incognita, though one should mention some special results recently obtained by Faber in [Fab13a] and [Fab13b]. In this project we obtain a relatively complete description of the combinatorial structure of an arbitrary finite morphism f : Y → X between smooth Berkovich curves, and it is divided into two parts.

Survey on the geometric Bogomolov conjecture

This is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory over function fields and a quick review on basic notions on non-archimedean analytic geometry.

Derived non-archimedean analytic spaces

We propose a derived version of non-archimedean analytic geometry. Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings. Such a naive definition turns out to be insufficient. In this paper, we resort to the theory of pregeometries and structured topoi introduced by Jacob Lurie. We prove the following three fundamental properties of derived non-archimedean analytic spaces: (1) The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the \(\infty \)-category of derived non-archimedean analytic spaces. (2) The \(\infty \)-category of derived non-archimedean analytic spaces admits fiber products. (3) The \(\infty \)-category of higher non-archimedean analytic Deligne–Mumford stacks embeds fully faithfully into the \(\infty \)-category of derived non-archimedean analytic spaces. The essential image of this embedding is spanned by n-localic discrete derived non-archimedean analytic spaces. We will further develop the theory of derived non-archimedean analytic geometry in our subsequent works. Our motivations mainly come from intersection theory, enumerative geometry and mirror symmetry.

Non-Archimedean analytic geometry as relative algebraic geometry

We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of To\"{e}n--Vaqui\'{e}--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry and in this way we also get definitions of (higher) non-Archimedean analytic stacks. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the category of varieties in our version of relative algebraic geometry. We also include a treatment of quasi-coherent sheaf theory in analytic geometry. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

Higher analytic stacks and GAGA theorems

We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert's theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre's GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-étale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special case.

Gromov compactness in non-archimedean analytic geometry

Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

Non-Archimedean Analytic Geometry

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to arithmetic and algebraic geometry. These applications include questions in Arakelov theory, p-adic differential equations, p-adic Hodge theory and the geometry of moduli spaces. Various methods were used in combination with analytic geometry, in particular perfectoid spaces, model theory, skeleta, formal geometry and tropical geometry. Mathematics Subject Classification (2010): 03C98, 11G25, 12H25, 14F20, 14G20, 14G22, 32P05. Introduction by the Organisers The half-size workshop Non-Archimedean Analytic Geometry, organized by Vladimir Berkovich (Rehovot), Walter Gubler (Regensburg) and Annette Werner (Frankfurt) had 26 participants. Non-Archimedean analytic geometry is a central area of arithmetic geometry. The first analytic spaces over fields with a nonArchimedean absolute value were introduced by John Tate and explored by many other mathematicians. They have found numerous applications to problems in number theory and algebraic geometry. In the 1990s, Vladimir Berkovich initiated a different approach to non-Archimedean analytic geometry, providing spaces with good topological properties which behave similarly as complex analytic spaces. Independently, Roland Huber developed a similar theory of adic spaces. Recent years have seen a growing interest in such spaces since they have been used to solve several deep questions in arithmetic geometry. We had 19 talks in this workshop reporting on recent progress in non-Archimedean analytic geometry and its applications. All talks were followed by lively 3212 Oberwolfach Report 53/2012 discussions. Several participants explained work in progress. The workshop provided a useful platform to discuss these new developments with other experts. During the workshop, we saw applications to complex singularity theory and to Brill–Noether theory in algebraic geometry. Progress was made in the study of Berkovich spaces over Z, and they were used for an arithmetic Hodge index theorem with applications to the non-archimedean Calabi-Yau problem. An analog of complex differential geometry was developed on Berkovich spaces which allows us to describe non-archimedean Monge-Ampere measures as a top-dimensional wedge product of first Chern forms or currents. Two talks focused on p-adic differential equations where Berkovich spaces help to understand the behaviour of radii of convergence. Scholze’s perfectoid spaces, which have led to spectacular progress regarding the monodromy weight conjecture, and their relations to padic Hodge theory were the topic of two other lectures. Methods from Model Theory become increasingly important in arithmetics, and we have seen two talks adressing this in connection with analytic spaces. Skeleta and tropical varieties are combinatorial pictures of Berkovich spaces, and these tools were used in several talks. In the one-dimensional case these methods lead to a better understanding of well-studied objects of algebraic geometry such as moduli spaces of curves or component groups. Non-Archimedean Analytic Geometry 3213 Workshop: Non-Archimedean Analytic Geometry

Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings

AbstractIn the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building$\mathcal{B}(\mathrm{G},k)$of a connected and reductive linear algebraic group G over a non-Archimedean fieldkin the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding$\mathcal{B}(\mathrm{G},k)$in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels

V.G. Berkovich’s non-Archimedean analytic geometry provides a natural framework to understand the combinatorial aspects in the theory of toric varieties and toroidal embeddings. This point of view leads to a conceptual and elementary proof of the following results: if X is an algebraic scheme over a perfect field and if D is the exceptional normal crossing divisor of a resolution of the singularities of X, the homotopy type of the incidence complex of D is an invariant of X. This is a generalization of a theorem due to D. Stepanov.

Local monodromy in non-Archimedean analytic geometry

Local monodromy in non-Archimedean analytic geometry

A new proof of the Gerritzen-Grauert theorem

The Gerritzen-Grauert theorem ([GG], [BGR, 7.3.5/1]) is one of the most important foundational results of rigid analytic geometry. It describes so called locally closed immersions between affinoid varieties, and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains. In its turn, the latter fact allowed one to extend Tate’s theorem (see [Tate], [BGR, 8.2.1/1]) on acyclicity of the Cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains. The same fact also plays an important role in foundations of non-Archimedean analytic geometry developed by V. Berkovich in [Ber1] and [Ber2]. Recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry (the latter were called in [Ber1] strictly affinoid) and, besides, the valuation on the ground field is not assumed to be nontrivial. In the recent papers by A. Ducros [Duc, 2.4] and the author [Tem, 3.5], the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces, but its proof was based on the case of strictly affinoid ones (i.e., affinoid varieties). The original proof of the Gerritzen-Grauert theorem is not easy, and since then the only different proof was found by M. Raynaud in the framework of his approach to rigid analytic geometry (see [Ray], [BL]). Although that proof is more conceptual, it is based on a complicated algebraic technics. The purpose of this paper is to give a new proof of the Gerritzen-Grauert theorem which uses basic properties of affinoid algebras in a standard way. The only novelty is in using the whole spectrum M(A) of an affinoid algebra A, introduced in [Ber1], instead of the maximal spectrum Max(A), considered in rigid analytic geometry. The use of the whole spectrum allows one to apply additional but standard compactness arguments. In §§1-2, we work in the setting of rigid analytic geometry, i.e., the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered. In §1, we recall basic definitions of an affinoid algebra, an affinoid domain, all notions necessary for the formulation of the Gerritzen-Grauert theorem, and formulate it (Theorem 1.1). The only new fact is Proposition 1.2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a

Vanishing cycles for non-archimedean analytic spaces

In this work we develop a formalism of vanishing cycles for non-Archimedean analytic spaces which is an analog of that for complex analytic spaces from [SGA7], Exp. XIV. As an application we prove that in the equicharacteristic case the stalks of the vanishing cycles sheaves of a scheme X at a closed point x E X, depend only on the formal completion Spf(Ox,,) of X at x. In particular, any continuous homomorphism OX,x -+ y,y induces a homorphism from the stalks of the vanishing cycles sheaves of X at x to those of Y at y. Furthermore, we prove that, given Ox,x and O y, , there exists n > 1 such that, for any pair of continuous homomorphisms OX,x -Oy,y that coincide modulo the n-th power of the maximal ideal of Oy,y, the induced homomorphisms between the stalks of the vanishing cycles sheaves coincide. These facts generalize a result of G. Laumon from [Lau] (see Remark 7.6). Throughout the paper we fix a non-Archimedean field k (whose valuation is not assumed to be nontrivial). In ?1 we study etale Galois sheaves on k-analytic spaces. To define the vanishing cycles functor and to work with it, we use the language of pro-analytic spaces, i.e., pro-objects of the category of analytic spaces ([SGA4], Exp. I). Examples of such objects are the germs of analytic spaces as in [Ber2], ?3.4. Another example is considered in ?3. In ?4 we define the vanishing cycles functor and establish its basic properties. In ?5 we show that the vanishing cycles sheaves are trivial for smooth morphisms. In ?6 we prove a comparison theorem for vanishing cycles. This theorem is more general than its analog over C from [SGA7], Exp. XIV, and its proof does not use Hironaka's theorem on resolution of singularities. In ?7 we apply the comparison theorem to prove the properties of the vanishing cycles sheaves of schemes formulated above. It is worthwhile to note that this application is obtained by considering non-Archimedean analytic geometry over fields with trivial valuation. Like [Ber3], this work arose from a suggestion of P. Deligne to apply the etale cohomology theory from [Ber2] to the study of the vanishing cycles sheaves of schemes. I am very grateful to him for useful discussions on the subject. I also