Vladimir G. Berkovich

Spectral theory and analytic geometry over non-Archimedean fields

The spectrum of a commutative Banach ring Affinoid spaces Analytic spaces Analytic curves Analytic groups and buildings The homotopy type of certain analytic spaces Spectral theory Perturbation theory The dimension of a Banach algebra.

Étale cohomology for non-Archimedean analytic spaces

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On the comparison theorem for étale cohomology of non-Archimedean analytic spaces

Let ϕ:Y →X be a morphism of finite type between schemes of locally finite type over a non-Archimedean fieldk, and letF be an étale constructible sheaf onY. In [Ber2] we proved that if the torsion orders ofF are prime to the characteristic of the residue field ofk then the canonical homomorphisms (RQϱ*F)an →Rqϱ*anFan are isomorphisms. In this paper we extend the above result to the class of sheavesF with torsion orders prime to the characteristic ofk.

A Non-Archimedean Interpretation of the Weight Zero Subspaces of Limit Mixed Hodge Structures

Let \(\mathcal {X}\) be a proper scheme over the field F of functions meromorphic in an open neighborhood of zero in the complex plane. The scheme \(\mathcal {X}\) gives rise to a proper morphism of complex analytic spaces \({\mathcal X}^h \to D^\ast\) and, if the radius of the open disc D is sufficiently small, the cohomology groups of the fibers \({\mathcal X}^h_t\) at points \(t\in D^\ast\) form a variation of mixed Hodge structures on \(D^\ast\), which admits a limit mixed Hodge structure. The purpose of the paper is to construct a canonical isomorphism between the weight zero subspace of this limit mixed Hodge structure and the rational cohomology group of the non-Archimedean analytic space \({\mathcal X}^{\rm an}\) associated to the scheme \(\mathcal {X}\) over the completion of the field F.

An analog of Tate's conjecture over local and finitely generated fields

Let K be a field complete with respect to a non-Archimedean valuation (which is not assumed to be nontrivial), and let X be a separated scheme of finite type over K. Recall that the K-analytic space Xan in the sense of [Ber1] and [Ber2], which is associated with X, is locally compact, countable at infinity, and locally arc-wise connected. Furthermore, Xan is compact if and only if X is proper; it is arc-wise connected if and only if X is connected, and the topological dimension of Xan is equal to the dimension of X. In [Ber7], it was proven that Xan is locally contractible if X is smooth and the valuation on K is nontrivial. In the course of the proof, the homotopy type of Xan is described for a broad class of schemes, and one of the consequences of that description (see [Ber7, Theorem 10.1]) states that the cohomology groups Hi(|Xan|,Z) are always finitely generated and that there exists a finite separable extension K ′ of K such that for any non-Archimedean field K ′′ over K ′, one has H(|(X ⊗ K ′)an|,Z) →̃ H(|(X ⊗ K ′′)an|,Z). Moreover, the same facts are also true for the cohomology groups with compact support Hc(|X an|,Z). Here, |X| denotes the underlying topological space of a K-analytic space X. Recall that, if the valuation on K is nontrivial, the above cohomology groups (without support) coincide with the cohomology groups of the associated rigid analytic space (see [Ber2, §1.6]). Notice also that for any abelian group A flat over Z, one has H(|X|,Z) ⊗ A →̃H(|X|, A) and Hc(|X|,Z)⊗A →̃Hc(|X|, A). Assume now that K is one of the following fields: (a) a local non-Archimedean field, (b) a field finitely generated over Z (i.e., it is finitely generated as a field over the

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 Hironakas 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

Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields, in particular to number theory and algebraic geometry. These applications included Mirror Symmetry, the Langlands program, p-adic Hodge theory, tropical geometry, resolution of singularities and the geometry of moduli spaces. Much emphasis was put on making the list of talks to reflect this diversity, thereby fostering the mutual inspiration which comes from such interactions. Mathematics Subject Classification (2010): 03 C 98, 11 G 20, 11 G 25, 14 F 20, 14 G 20, 14 G

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