La droite de Berkovich sur Z

Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables

En nous appuyant sur une construction due à Bedford et Taylor, et certains résultats récents de Demailly, nous présentons une extension (partielle) de la géométrie d'Arakelov aux fibrés en droites intégrables. (Ces derniers sont les fibrés en droites hermitiens sur une variété arithmétique $X$ pouvant se décomposer sous la forme $\ov{E} = \ov{E}_{1}\otimes (\ov{E}_{2})^{-1}$, où $\ov{E}_{1}$ et $\ov{E}_{2}$ sont des fibrés en droites munis à l'infini d'une métrique continue approchable uniformément sur $X(C)$ par des métriques positives $C^{\infty}$). Nous appliquons notre théorie aux fibrés e…

Normes de droites sur les surfaces cubiques

Let $k$ be a field and $X \subset P^3_{k}$ a smooth cubic surface. Let $Δ\subset Pic(X)$ be the finite index subgroup spanned by norms of lines on $X_{K}$ for $K$ running through the finite separable extensions of $k$. The quotient $Pic(X)/Δ$ is 3-primary. If $X$ contains a line defined over $k$, then $Δ=Pic(X)$.

Cohomologie des fibrés en droites sur les compactifications des groupes réductifs

On s'intéresse aux compactifications équivariantes d'un groupe réductif quelconque, $G$, vu comme espace homogène pour $G \times G$. Pour un revêtement fini $\widetilde{G}$ de $G$, les groupes de cohomologie des fibrés en droites sur ces compactifications sont naturellement des $\widetilde{G} \times \widetilde{G}-$modules de dimension finie. On détermine, dans cet article, tous ces $\widetilde{G} \times \widetilde{G}-$modules en calculant leur multiplicité selon chaque $\widetilde{G} \times \widetilde{G}-$module simple. La formule obtenue est en particulier valable pour les compactifications m…

Espaces de Berkovich sur $\mathbb{Z}$: morphismes étales

We develop properties of unramified, étale and smooth morphisms between Berkovich spaces over $\mathbb{Z}$. We prove that they satisfy properties analogous to those of morphisms of schemes and we provide analytification criteria. Our results hold for any valued field, rings of integers of a number field and discrete valuation rings. Those cases are treated by a unified way.

Drinfeld上半平面上等变线丛的模$p$上同调

Cohomologie mod $p$ des fibrés en droites équivariants sur le demi-plan de Drinfeld

该研究分类了Drinfeld上半平面半稳定模型上的等变线丛，研究其特殊纤维上线上丛的模p上同调，建立了一类权-1正线丛与不可约超奇异表示的一一对应关系。

Espaces de Berkovich sur Z : étude locale

We investigate the local properties of Berkovich spaces over Z. Using Weierstrass theorems, we prove that the local rings of those spaces are noetherian, regular in the case of affine spaces and excellent. We also show that the structure sheaf is coherent. Our methods work over other base rings (valued fields, discrete valuation rings, rings of integers of number fields, etc.) and provide a unified treatment of complex and p-adic spaces.

Raccord sur les espaces de Berkovich

Let $X$ be a Berkovich space over a valued field. We prove that every finite group is a Galois group over $\Ms(B)(T)$, where $\Ms(B)$ is the field of meromorphic functions over a part $B$ of $X$ satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over $K(T)$, where $K$ is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over ${\bf Z}$ and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite group is a Galois group over a field of convergent arithmetic power serie…

Sur la cohomologie à support des fibrés en droites sur les variétés symétriques complètes

Let $X$ be a complete symmetric variety i.e. the wonderful compactification of a symmetric $G-$homogeneous space (where $G$ is a simply-connected semi-simple linear algebraic group). If $L$ is a line bundle over $X$ and if $C$ is a Bialynicki-Birula cell of codimension $c$ in $X$, then the Lie algebra $\mathfrak g$ of $G$ operates naturally on the cohomology group with support : $H^c_C(L)$. One gives here a necessary condition on the cell $C$ for that $\mathfrak g-$module have a finite dimensional simple subquotient. As applications one calculates the Euler-Poincaré characteristic of $L$ over …

Berkovich 空间上的实微分形式与流

Formes différentielles réelles et courants sur les espaces de Berkovich

本文在 Berkovich 空间上建立实 (p,q)-形式与流理论，推广积分、Stokes、Poincaré-Lelong、Bedford-Taylor 及 Chern 相交工具，并以热带空间框架重释热带相交理论。

A Poincaré lemma for real-valued differential forms on Berkovich spaces

Real-valued differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros in 'Formes différentielles réelles et courants sur les espaces de Berkovich' using superforms on polyhedral complexes. We prove a Poincaré lemma for these superforms and use it to also prove a Poincaré lemma for real-valued differential forms on Berkovich spaces. For superforms we further show finite dimensionality for the associated de Rham cohomology on polyhedral complexes in all (bi-)degrees. We also show finite dimensionality for the real-valued de Rham cohomology of the analytification…

Normalized Berkovich spaces and surface singularities

We define normalized versions of Berkovich spaces over a trivially valued field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by rescaling semivaluations. We associate such a normalized space to any special formal $k$-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed $G$-topological space, which we prove to be $G$-locally isomorphic to a Berkovich space over the field $k((t))$ with a $t$-adic valuation. These spaces can be interpreted as non-archimedean models for the…

About Hrushovski and Loeser's work on the homotopy type of Berkovich spaces

Those are the notes of the two talks I gave in april 2013 in St-John (US Virgin Islands) during the Simons Symposium on non-Archimedean and tropical geometry. They essentially consist of a survey of Hrushovski and Loeser's work on the homotopy type of Berkovich spaces; the last section explains how the author has used their work for studying pre-image of skeleta.

在混合紧致化上的群作用

Action de groupe sur la compactification hybride

本文研究了在混合紧致化上群作用的扩展，通过刻画良定义的集合U，展示了其商空间与紧致化同胚，并应用于有理映射空间和SL_2群，得到更一般的结果。

Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry

This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.

Descent for non-archimedean analytic spaces

In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the second type, including the descent of properties of being a good analytic space and being a morphism without boundary.