Altered local uniformization of Berkovich spaces

Metric uniformization of morphisms of Berkovich curves via $p$-adic differential equations

We consider a finite étale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton $Γ_f=(Γ_Y,Γ_X)$ of the morphism $f$. We prove that $Γ_f$ radializes $f$ if and only if $Γ_X$ controls the pushforward of the constant $p$-adic differential equation $f_*(\mathcal{O}_Y,d_Y)$. Furthermore, when $f$ is a finite étale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\in X$, counted without multiplicity, only depe…

Berkovich curves and Schottky uniformization

This text is an exposition of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It consists of two parts, the first one of an introductory nature, and the second one more advanced. The first part is meant to be an introduction to the theory of Berkovich spaces focused on the case of the affine line. We define the Berkovich affine line and present its main properties, with many details: classification of points, path-connectedness, metric structure, variation of rational functions, etc. Contrary to many other introductory texts, we do not assume th…

Metric uniformization of morphisms of Berkovich curves

We show that the metric structure of morphisms $f\colon Y\to X$ between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton $Γ=(Γ_Y,Γ_X)$ of $f$, the sets $N_{f,\ge n}$ of points of $Y$ of multiplicity at least $n$ in the fiber are radial around $Γ_Y$ with the radius changing piecewise monomially along $Γ_Y$. In this case, for any interval $l=[z,y]\subset Y$ connecting a rigid point $z$ to the skeleton, the restriction $f|_l$ gives rise to a $profile$ piecewise monomial function $φ_y\colo…

簇与非阿基米德空间的局部一致化的高度约化

Height reduction for local uniformization of varieties and non-archimedean spaces

本文证明概形与非阿基米德解析空间的局部一致化可严格归结为高度为一的赋值情形，从而将非阿基米德空间的问题转化为研究光滑Berkovich空间的局部结构。

Altered Local Uniformization Of Rigid-Analytic Spaces

We prove a version of Temkin's local altered uniformization theorem. We show that for any rig-smooth, quasi-compact and quasi-separated admissible formal $\mathcal{O}_K$-model $\mathfrak{X}$, there is a finite extension $K'/K$ such that $\mathfrak{X}_{\mathcal{O}_{K'}}$ locally admits a rig-étale morphism $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and a rig-isomorphism $h\colon \mathfrak{X}" \to \mathfrak{X}'$ with $\mathfrak{X}'$ being a successive semi-stable curve fibration over $\mathcal{O}_{K'}$ and $\mathfrak{X}"$ being a poly-stable formal $\mathcal{O}_{K'}$-scheme. Mor…

Dominant classes of projective varieties

We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.