Questions tagged [berkovich-geometry]
The berkovich-geometry tag has no usage guidance.
71 questions
3
votes
0
answers
169
views
What are non-archimedean norms on $\mathbb{R}$, whose restriction to $\mathbb{Q}$ is trivial?
I wonder if there is any classification result on non-archimedean norms on $\mathbb{R}$, with trivial restriction to $\mathbb{Q}$? Any references or examples would be welcomed!
Some examples of such ...
- 241
0
votes
0
answers
128
views
Notion of Kahler differentials for Berkovic spaces
What is, in abstract analytic geometry (I mean, for example, in Berkovic spaces), the approach used for differential forms?
Ordinary Kahler differentials from commutative algebra/algebraic geometry ...
- 101
3
votes
0
answers
91
views
Quillen-Suslin theorem for non-strict polydiscs in the sense of Berkovich
Let $K$ be a complete non-archimedean field of mixed characteristic $(0,p)$. Choose $\rho_1,\dots,\rho_n\in \mathbb{R}_{>0}$ and let $P$ be a finite projective module over $K\langle\rho_1^{-1}t_1,\...
- 63
7
votes
1
answer
361
views
Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin–Verdier site?
Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...
- 804
3
votes
1
answer
77
views
Connected Berkovich affinoid space with unique Shilov boundary
Let $X$ be a smooth connected affinoid space over a (algebraic closed, spherically complete) nonarchimedean field $K$, and $X’$ be an affinoid subspace contained in the interior of $X$, with Shilov ...
- 63
16
votes
2
answers
941
views
Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?
I was reading the proof of Ostrowski's theorem, with an eye toward the Zariski-Riemann space (as well as adic space, Berkovich space, etc.) In the proof, the value group is always assumed to be in $\...
- 343
1
vote
1
answer
268
views
Formal series which are always zero
Let $(k, |\cdot|)$ be a complete field with a non-Archimedean norm, not necessarily algebraically closed. Define the Tate algebra as follows:
\begin{align*}
k \langle T_1, \dots, T_n \rangle = \{ \...
- 1,485
2
votes
0
answers
269
views
Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?
As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...
- 804
4
votes
0
answers
251
views
What information does the topology of nonarchimedean Berkovich analytic spaces encode?
Given a finite type scheme $X$ over $\Bbb{C}$ we can associate to it an analytic space $X^\text{an}$. There are then comparison theorems comparing invariants of the topological space $X^\text{an}$ ...
- 193
1
vote
0
answers
183
views
Contractibility of the quotient of an analytification of a smooth variety by a finite group (if the field is trivially valued)
Let $k$ be a field and $X$ be a smooth irreducible $k$-variety with an action of a finite group $G$. I consider $k$ as a trivially valued field.
It is known from results of Berkovich ("Smooth p-...
- 41
3
votes
1
answer
502
views
"Non-algebraic" Berkovich spaces
Usually, Berkovich analytic spaces are derived from some Banach rings (or chains of Banach rings) over a completely normed field $k$ through Berkovich spectrum. But when the base field is the complex ...
- 1,543
24
votes
1
answer
2k
views
Why are there three kinds of non-archimedean geometry?
It may seem silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to ...
- 1,270
6
votes
1
answer
242
views
Why are Berkovich spaces locally connected?
A characteristic feature of Berkovich spaces is that they are locally connected (in fact, locally contractible). I'd like to understand the proof. The key ingredient seems to be Corollary 2.2.8 in ...
- 63.9k
5
votes
1
answer
178
views
An example where the non-Archimedean tensor product of normed modules is only seminormed?
Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies
$$ | 0_R| = ...
- 153
2
votes
0
answers
248
views
Enlightening examples of tropical skeletons of Berkovich spaces
Let $K$ be a complete non-archimedean field and let $X$ be a $K$-analytic space in the sense of Berkovich of pure dimension $d$. Let $\varphi \colon X \to \mathbf{G}_m^r$ be a moment map to an ...
user478030
1
vote
1
answer
209
views
Reference request: Gruson's theorem on the tensor product of Banach spaces over a non-Archimedean field
I am looking for a reference for theorem 3.21 of these notes: https://web.math.princeton.edu/~takumim/Berkovich.pdf
The theorem states that if $k$ is a non-Archimedean field and $X$ and $Y$ are $k$-...
- 63
7
votes
0
answers
334
views
Berkovich spaces — why use atlases?
Note: this question only concerns Berkovich spaces ("analytic spaces") as defined in Berkovich's first book "Spectral Theory and Analytic Geometry over Non-Archimedean Fields", not ...
- 28.2k
45
votes
2
answers
4k
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-...
- 28.2k
7
votes
1
answer
325
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 ...
- 998
2
votes
1
answer
300
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 $\...
- 1,153
5
votes
0
answers
201
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 ...
- 445
4
votes
1
answer
392
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 ...
- 1,838
5
votes
0
answers
194
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 ...
- 5,901
5
votes
1
answer
574
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,...
- 4,398
2
votes
0
answers
125
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 ...
- 21
4
votes
1
answer
255
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 ...
- 1,153
1
vote
0
answers
177
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 ...
- 4,398
3
votes
0
answers
183
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
votes
1
answer
976
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 ...
- 332
8
votes
1
answer
384
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 ...
- 1,153
6
votes
1
answer
670
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. ...
- 1,153
1
vote
0
answers
80
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 ...
- 2,789
3
votes
0
answers
228
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 ...
- 1,153
3
votes
0
answers
105
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 ...
user140765
1
vote
0
answers
131
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$....
- 2,789
1
vote
0
answers
255
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-...
- 2,789
4
votes
1
answer
480
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$...
- 2,789
2
votes
1
answer
139
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....
- 155
3
votes
1
answer
308
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/...
- 155
7
votes
1
answer
452
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 ...
- 1,790
7
votes
1
answer
367
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 ...
- 2,789
13
votes
2
answers
1k
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 ...
- 2,789
5
votes
0
answers
187
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 $\...
- 892
4
votes
1
answer
109
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....
- 4,070
5
votes
1
answer
314
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 (|\...
- 81
9
votes
0
answers
267
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
...
- 4,070
4
votes
0
answers
150
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 $\{a_i\} \in D(a,r)$ ...
- 61
6
votes
2
answers
323
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 ...
- 4,070
1
vote
0
answers
146
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 ...
- 4,070
13
votes
0
answers
496
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 ...
- 5,053