All Questions
Tagged with berkovich-geometry rigid-analytic-geometry
14 questions with no upvoted or accepted answers
17
votes
0
answers
952
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 ...
- 171
10
votes
0
answers
409
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 $\...
- 582
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
7
votes
0
answers
672
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 ...
- 71
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
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)...
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
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
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
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 ...
- 4,408
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
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