Questions tagged [berkovich-geometry]

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3
votes
0answers
97 views

Is there a notion of Frobenius weight on the étale cohomology of Berkovich analytic spaces?

When dealing with the étale cohomology groups $\mathrm H_c^i(X\otimes \overline{\mathbb F_q},\overline{\mathbb Q_{\ell}})$ of a variety $X$ over $\mathbb F_q$ (where $q$ is a power of some prime ...
32
votes
2answers
2k views

Are rigid-analytic spaces obsolete, since adic spaces exist?

Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer: Are there any good reasons to use Tate's theory of rigid-...
2
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0answers
170 views

Metrics on vector bundles over Berkovich analytic spaces

$\newcommand{\an}{{\mathrm{an}}} \newcommand{\blank}{{-}} \newcommand{\sheaf}{\mathcal} \newcommand{\comp}{\widehat} \newcommand{\from}{\colon} \newcommand{\IR}{\mathbb{R}} \newcommand{\Spec}{\mathrm{...
7
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1answer
173 views

Indeterminacy locus of meromorphic maps of rigid analytic spaces

Setup. Let $k$ be an algebraically closed field of characteristic zero. Let $X/k$ be a normal variety, and let $Y/k$ be a proper variety. It is well-known that the indeterminacy locus of a rational ...
2
votes
1answer
168 views

Is $\mathbb{A}_k^n(k)$ dense in the Berkovich analytification of $\mathbb{A}_k^n$?

Let $k$ be a non-archimedean field and denote by $\mathbb{A}_k^n$ the analytic affine space of $n$ dimensions over $k$ (analytic in the sense of Berkovich). There is a natural injective map of sets $\...
5
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0answers
113 views

Berkovich Integration on algebraic curves

Berkovich developed a theory of integrating one-forms on his analytic spaces in his book "Integration of One-forms on $P$-adic analytic spaces". As this book is difficult to digest for me, I ...
4
votes
1answer
256 views

Topological and algebraic covering spaces in Berkovich geometry

Let $k$ be a complete, non-archimedean field, and $X$ a Berkovich space over $k$ (as nice as you like, for arguments sake let's say strictly $k$-analytic, good, and geometrically connected). As ...
4
votes
0answers
122 views

Deformations of vector bundles and tubular neighborhood

I had a number of questions that are somewhat related to each other. I decided to post them altogether instead of separately. I'd appreciate any kinds of answers, ideas or sources regarding any of ...
4
votes
1answer
256 views

Identity theorem in $p$-adic geometry/analysis

If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $\mathbb{Q}_p$ is disconnected. The modern approach to this is,...
2
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0answers
91 views

A structure sheaf for real analytification of semialgebraic sets in the context of signed tropicalization

Let $X=Spec(A)$ be an affine scheme, where $A$ be a commutative algebra over a non-archimedean valued field $K$. Assume that $K$ is a real closed field with the unique ordering $<$, which should be ...
4
votes
1answer
183 views

What is the definable functor associated to an algebraic scheme (model theory of valued fields)

I have a very basic question regarding algebraic model theory. I am trying to read Espaces de Berkovich, polytopes, squelettes et théorie des modèles (MSN) by Antoine Ducros. The relevant section is ...
1
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0answers
116 views

L-function in p-adic spaces

I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...
3
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0answers
141 views

gluing Berkovich spaces

In his paper Etale cohomology for non-Archimedean analytic space (IHES), Berkovich explained how to glue $k$-analytic spaces (Prop. 1.3.3) and show its uniqueness using the Prop 1.3.2 (gluing morphism)...
9
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1answer
534 views

Completed tensor product is exact

In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...
7
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1answer
238 views

Are maps corresponding to affinoid subdomains flat in the Banach sense?

$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$ Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid ...
6
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1answer
399 views

An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$

Let $k$ be a complete non-archimedean field and let $\varphi \colon X \to Y$ be a morphism of rigid analytic spaces over $k$, where $\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$ is affinoid. ...
1
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0answers
54 views

Polytopal domains in non-archimedean torus

Given a non-archimedean field $\mathbb K$, there is a natural map $$ \mathrm{val}: (\mathbb K^*)^n\to\mathbb R^n$$ (See Section 4 of Gubler's paper). Gubler mentions there $\mathrm{val}$ is a ...
3
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0answers
199 views

Is the Čech complex of a coherent sheaf on a compact separated rigid analytic space admissible?

$\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$Let $X$ be a compact, separated rigid $k$-analytic space over some complete non-archimedean field $k$. Then $X$ has a finite affinoid covering ...
3
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0answers
96 views

A proper analytic surface into which every smooth proper analytic curve embeds

Let $k$ be a finite extension of $\mathbb{Q}_p$. Does there exist a proper $k$-analytic surface such that there is a closed immersion into it from any connected smooth proper $k$-analytic curve? The ...
1
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0answers
106 views

Affinoid algebra and fundamental theorem of algebra

This post is closely related to the previous one here. But more generally, we want to study an affinoid algebra $A:=T_n/\mathfrak a$. Let's assume $\mathfrak a= (f_1,\dots,f_r)$ for some $f_i\in T_n$....
1
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0answers
141 views

Explicit description of rigid analytification of torus

It is known that in non-archimedean world there is also a GAGA-functor from the category of $K$-schemes of locally finite type to the category of rigid $K$-spaces. Here $K$ is a field with a non-...
3
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1answer
342 views

Tate algebras and fundamental theorem of algebra

Let $\mathbb K$ be an algebraically-closed complete non-archimedean field whose absolute value is non-trivial. Consider the Tate algebra $T_n=\mathbb K\langle X_1,\dots, X_n \rangle$ and fix $f\in T_n$...
2
votes
1answer
123 views

Definition of model functions and their density in $C^0(X^\text{an})$

I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10....
3
votes
1answer
201 views

Definition of a vertical ideal sheaf and a vertical fractional ideal sheaf

I'm working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/...
7
votes
1answer
289 views

The weight filtration on etale cohomology and Berkovich analytic geometry

If $X$ is a smooth projective curve over $\mathbb C_p$, then its first etale cohomology $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W_\bullet$ -- also ...
7
votes
1answer
308 views

Formal power series in Berkovich geometry

In this Berkovich's paper, the following kind of algebra is studied: $$ A=A_{m,n}=k^\circ \langle T_1,\dots,T_m \rangle [[S_1,\dots,S_n]] $$ where $k$ is some non-archimedean field with non-trivial ...
13
votes
2answers
717 views

Berkovich space including both archimedean and non-archimedean worlds

From this Temkin's paper (at the end of section 1.1.3), I know that one may define Berkovich spaces that include both archimedean and non-archimedean worlds. This looks very interesting. Temkin ...
5
votes
0answers
129 views

Ring of functions of generic fiber of affine special formal schemes

Fix $R$ a complete DVR. Recall from Berkovich's Vanishing Cycles for Formal Schemes II paper that we have a class of special formal schemes which are not topologically of finite type over $\...
4
votes
1answer
97 views

finite number of vertices of the polyhedron of variation of an invertible function on a Berkovich curve

The paper of Ducros "Cohomologie non-ramifiée sur une courbe p-adique lisse" mentions a theorem (1.21) about the existence of a "polyhedron of variation" of an invertible function on a Berkovich curve....
5
votes
1answer
271 views

projection to the scheme

Let $A$ be an $k$-affinoid algebra. Let $X=M(A)$ be the affinoid space associated to $A$ in the sense of Berkovich. There exists a natural morphism $\pi:X\to Y:=Spec(A)$ which sends $x$ to $ker (|\...
9
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0answers
218 views

Kodaira embedding theorem for rigid analytic varieties

Kodaira embedding theorem can be regarded as a vast generalizaton of the projectivity criterion for complex tori: indeed, the Riemann conditions essentially say that the line bundle defined by the ...
2
votes
0answers
172 views

generic points of Berkovich spaces and generic properties

Let $k$ be a field equipped with non-Archimedean absolute value, let $S=\mathcal{M}(A)$ be an affinoid domain over $k$, and let $\pi_S: S^{an} \to \tilde{S}$ be the reduction map from the Berkovich ...
4
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0answers
135 views

The image of annuli of the non-Archimedean projective line by rational functions

I'm reading the book "potential theory and dynamics over the Berkovich projective line" by Baker and Rumely. The proposition 2.18 in this claims that if you choose suitable finite $\{a_i\} \in D(a,r)$ ...
6
votes
2answers
271 views

Abel-Jacobi map for Mumford curves analytically

Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL_2(K)$, and let $X_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid ...
1
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0answers
130 views

characterization of the subspace of the moduli space of curves with maximally degenerate Jacobian

Let $K$ be a field equipped with a non-Archimedean absolute value, for example $K=\mathbb{C}((t))$. An Abelian variety $A$ over $K$ is called maximally degenerate if it admits an analytic ...
13
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0answers
454 views

Strange formula in arithmetic dynamic

Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two. We discovered the following operator which acts on the space of polynomials (or ...
28
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2answers
1k views

Is there a geometric realization of $\mathbf{C}((t))$-varieties?

Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle. When $F = \...
8
votes
1answer
371 views

Good analytic spaces over a field into locally ringed spaces is fully faithful

Let $k$ be a field which is complete with respect to a non-trivial non-archimedean rank-1 valuation, and let $X$ be scheme which is locally of finite type over $k$. In section of 3.5 of Berkovich's ...
9
votes
1answer
499 views

Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?

Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum $$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$ Now ...
7
votes
2answers
614 views

Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids

Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ ...
6
votes
1answer
345 views

Relations between two definitions of non-archimedean analytic spaces

I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature. Let us fix a non-archimedean complete valuation field $...
2
votes
2answers
476 views

Is there a notion of pure dimension for Berkovich analytic space?

For affinoid spaces the definition is similar to algebraic geometry, what about general analytic spaces? I can't find a reference about it. If yes then is the analytification of a variety of pure ...
1
vote
2answers
505 views

Berthelot functor, rigid analytic space

If $X=\operatorname{Spec} A$, where $A$ is a noetherien, complete local ring, with a finite residual field $\mathbb{F}_p$. We can associate to $A$ a rigid analytic space with two different ways, we ...
10
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0answers
332 views

Detecting $k$-affinoid spaces by vanishing cohomology

The property of being an affine scheme can be tested against all quasi-coherent sheaves in the following sense: a noetherian scheme $X$ is affine iff $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent $\...
5
votes
1answer
303 views

Berkovich stalk versus rigid analytic stalk

Let $A$ be a strictly affinoid algebra. Let $X^{Ber}$ bet its Berkovich spectrum and $X^{Tate} = \operatorname{Sp} A$ its affinoid variety in the sense of rigid analytic geometry. Let $\mathfrak{m} \...
10
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4answers
2k views

applications of Berkovich spaces

What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$
18
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2answers
2k views

Open problems in Berkovich geometry

I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...
2
votes
1answer
318 views

The tensor product of admissible morphisms of semi-normed modules over a normed ring is an admissible morphism (V. G. Berkovich)

Disclaimer : I found here https://mathoverflow.net/editing-help in the spoilers paragraph that putting >! would hide following things, which was a way for me to alleviate my question's presentation by ...
3
votes
1answer
411 views

is every point of a Berkovich space a Shilov point?

Let $k$ be an algebraically closed non-Archimedean valued field with the value group $\mathbb R$, and let $X$ be a variety over $k$. Is it true that for any point $x \in X^{an}$ of the Berkovich ...
2
votes
1answer
166 views

The target of a regular function in Non-archimedean analytic geometry

Let $(k,|\cdot|)$ be an algebraically closed field, complete wrt a (multiplicative) norm as in the framework of the Berkovich's analytic geometry. Given a commutative Banach $k$-algebra $\mathcal{A}\...