Le but de cette note est de donner une démonstration complète du théorème 4.1 de [5] qui a pour objet d’expliciter l’action de l’inertie modérée sur la semi-simplifiée modulo $p$ d’une certaine famille (assez restreinte) de représentations cristallines $V$ du groupe de Galois absolu d’un corps $p$-adique $K$. Lorsque $K$ n’est pas absolument ramifié, le calcul de cette action a déjà été accompli par Fontaine et Laffaille qui ont montré qu’elle est entièrement déterminée par les poids de Hodge-Tate de $V$, au...

Let $X$ be a smooth projective surface, $K$ the canonical divisor, $H$ a very ample divisor and $M_H(c_1,c_2)$ the moduli space of rank-two vector bundles, $H$-stable with Chern classes $c_1$ and $c_2$. We prove that, if there exists $c_1^{\text{'}}$ such that $c_1$ is numerically equivalent to $2c_1^{\text{'}}$ and if $c_2-\frac{1}{4}{c}_1^2$ is even, greater or equal to $H^2+HK+4$, then there is no Poincaré bundle on $M_H(c_1,c_2)\times X$. Conversely, if there exists $c_1^{\text{'}}$ such that the number $c_1^{\text{'}}·c_1$ is odd or if $\frac{1}{2}{c}_1^2-\frac{1}{2}{c}_1·K-c_2$ is odd, then there exists a Poincaré bundle on $M_H(c_1,c_2)\times X$.

Je présenterai des résultats de T. Ekedahl et H. Esnault sur les variétés projectives lisses sur un corps de caractéristique strictement positive, disons $p$, dont deux points peuvent être liés par une chaîne de courbes rationnelles, par exemple faiblement unirationnelles, ou de Fano. Notamment : 1) sur un corps fini, de telles variétés ont un point rationnel, résultat qui généralise le théorème de Chevalley-Warning ; 2) sur un corps algébriquement clos, de telles variétés ont un groupe fondamental...

We study the relationship between positivity of restriction of line bundles to general complete intersections and vanishing of their higher cohomology. As a result, we extend classical vanishing theorems of Kawamata–Viehweg and Fujita to possibly non-nef divisors.

We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles ${ℳ}_L$ that arise as the...

The pre-Tango structure is an ample invertible sheaf of locally exact differentials on a variety of positive characteristic. It is well known that pre-Tango structures on curves often induce pathological uniruled surfaces. We show that almost all pre-Tango structures on varieties induce higher-dimensional pathological uniruled varieties, and that each of these uniruled varieties also has a pre-Tango structure. For this purpose, we first consider the p-closed rational vector field induced...

We show that for a local, discretely valued field $F$, with residue characteristic $p$, and a variety $𝒰$ over $F$, the map $\rho :\mathrm{Gal}(F^{\mathrm{sep}}/F)\to \mathrm{Out}\left({\pi }_{1,\mathrm{geom}}^{\left(p^{\text{'}}\right)}\right(𝒰\left)\right)$ to the outer automorphisms of the prime to $p$ geometric étale fundamental group of $𝒰$ maps the wild inertia onto a finite image. We show that under favourable conditions $\rho$ depends only on the reduction of $𝒰$ modulo a power of the maximal ideal of $F$. The proofs make use of the theory of logarithmic schemes.