Related papers: Notes sur la droite projective de Berkovich
We develop in this article flattening techniques for coherent sheaves in the realm of Berkovich spaces; we are inspired by the general strategy that Raynaud and Gruson have used for dealing with the analogous problem in scheme theory. As an…
A. Tarski uses in his system for the elementary geometry only the primitive concept of point, and the two primitive relations betweenness and equidistance. Another approach is the relations to be on lines instead of points. W.…
We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the…
We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear) concerning restricting uniformly open, continuous surjections to smaller subspaces where they…
In a spherically complete ultrametric space, a strictly contracting mapping has a fixed point. We indicate in this paper how this fixed point can either be reached or approximated.
The purpose of the present article is to examine the essence of what has commonlybeen described as a "projective line", but which is here named a "meridian". This shall be done in several papers: this first paper devoted to the meridian…
We provide a projective description of the space $\mathcal{E}^{\{\mathfrak{M}\}}(\Omega)$ of ultradifferentiable functions of Roumieu type, where $\Omega$ is an arbitrary open set in $\mathbb{R}^d$ and $\mathfrak{M}$ is a weight matrix…
This paper extends our previous works arXiv:1802.07306 [math.NT], arXiv:1808.02382 [math.NT] on determining the spectrum, in the Berkovich sense, of ultrametric linear differential equations. Our previous works focused on equations with…
In these introductory notes I explain some basic ideas in string field theory. These include: the concept of a string field, the issue of background independence, the reason why minimal area metrics solve the problem of generating all…
Adapting the notion of the spectrum $\Sigma_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $\sigma^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$.…
The notion of quasi-unit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potential theoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned…
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge…
The aim of these lecture notes is, after having quickly described various compactifications of the Teichm\"{u}ller space of a compact connected oriented surface minus finitely many points, to give a construction, by the equivariant Gromov…
We revisit congruence zeta functions of smooth projective varieties over finite fields in the framework of Scholze's Berkovich motives. Via this formalism and categorical traces, we construct a new zeta function, and show that it agree with…
We give a geometric characterisation of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line…
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in…
We investigate the topological properties of Berkovich analytifications over hybrid fields, that is a field equipped with the maximum of its native norm and the trivial norm. We prove that the analytification of the affine line or of a…
These are some informal notes concerning topological vector spaces, with a brief overview of background material and basic notions, and emphasis on examples related to classical analysis.
Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebra A with a fixed projection p. The resulting space P(p) admits a rich geometrical structure as a…