Abstract Given a polarized projective variety ( X , L stretchy="false">) {(X,L)} over any non-Archimedean field, assuming continuity of envelopes, we define metric on the space finite-energy metrics L , related to construction Darvas in c...

Abstract We transfer several elementary geometric properties of rigid-analytic spaces to the world adic spaces, more precisely category which are locally (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and Stein factorization theorem. Most notably, we show that morphism in our is automatically open if target nor...

In this article, we show that the Goldman–Iwahori metric on space of all norms a fixed vector satisfies Helly property for balls. On non-Archimedean side, deduce most classical Bruhat–Tits buildings may be endowed with natural piecewise ? ? $\ell ^\infty$ which is injective. We also prove semisimple groups over local fields act properly and cocompactly graphs. This gives another proof biautomat...

‎hensel [k‎. ‎hensel‎, ‎deutsch‎. ‎math‎. ‎verein‎, ‎{6} (1897), ‎83-88.] discovered the $p$-adic number as a‎ ‎number theoretical analogue of power series in complex analysis‎. ‎fix ‎a prime number $p$‎. ‎for any nonzero rational number $x$‎, ‎there‎ ‎exists a unique integer $n_x inmathbb{z}$ such that $x = ‎frac{a}{b}p^{n_x}$‎, ‎where $a$ and $b$ are integers not divisible by ‎$p$‎. ‎then $|x...

This article has three major goals. First, we define tropical cycle class maps for smooth varieties over non-Archimedean fields, valued in the Dolbeault cohomology defined in terms of real forms introduced by Chambert-Loir and Ducros. Second, we construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over certain non-Archimedea...