# Questions tagged [berkovich-geometry]

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48
questions

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### 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

**1**answer

158 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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99 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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128 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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**1**answer

376 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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**1**answer

195 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**

votes

**1**answer

318 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. ...

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47 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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176 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 ...

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93 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 ...

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103 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 $f_i\in T_n$....

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120 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**

votes

**1**answer

319 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**

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**1**answer

117 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**

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**1**answer

172 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**

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226 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**

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**1**answer

285 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**

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**2**answers

634 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**

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123 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**

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**1**answer

93 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

**1**answer

270 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 (|\...

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207 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**

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164 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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129 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 $\{a_i\} \in D(a,r)$ ...

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247 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 ...

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127 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 ...

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436 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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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**

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**1**answer

341 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**

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**1**answer

478 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**

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547 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**

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326 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 $...

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420 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 ...

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442 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 ...

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322 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 $\...

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274 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} \...

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**4**answers

1k views

### applications of Berkovich spaces

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

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**2**answers

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 ...

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**1**answer

296 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**

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**1**answer

375 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 ...

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156 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}\...

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808 views

### A functor of points approach to Berkovich analytic spaces

Is it possible to define a Berkovich analytic space via its functor of points?
Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space ...

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537 views

### Etale cohomology of Berkovich spaces

Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the ...

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**1**answer

615 views

### How should we understand the relative interior in Berkovich spaces

I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of
\...

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3k views

### Rigid analytic spaces vs Berkovich spaces vs Formal schemes

I wonder if someone could explain briefly what is the relation between these 3 formal models, of a Berkovich space, a rigid analytic space and a formal scheme?
I have been working with formal schemes ...

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647 views

### Cohesive ∞-toposes for analytic geometry

There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes).
...

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3k views

### Higher dimensional berkovich spaces

I am looking for a geometric and topological way to make a visualization of higher dimensional berkovich spaces, starting with the berkovich plane. Of course, this is just a collection of bounded semi-...

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**1**answer

1k views

### Do Berkovich homogenous spaces exist?

Let G be a k-analytic group, and let H be a closed subgroup of G. Then does there exist a k-analytic space, which can be reasonably called the quotient G/H?
Commentary: I realise that I am not being ...