Cet article est le résultat d'une relecture de la démonstration de la conjecture {\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}faiblement admissible implique admissible{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)} par J.-M. Fontaine et l'auteur. On donne une version renforcée de l'un des ingrédients (le {\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}lemme fondamental{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)}) de cette démonstration, ce qui nous amène à introduire un corps non commutatif gigantesque $\mathfrak{C}$, de centre $\textbf{Q}_p$ et contenant un corps $C$ algébriquement clos et complet pour la norme $p$-adique comme sous-corps commutatif maximal. Un peu {\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}d'algèbre linéaire{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)} sur ce corps mène à l'introduction de la catégorie des Espaces (avec un E majuscule) de Banach de dimension finie, et la conjecture {\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}faiblement admissible implique admissible{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)} se réduit alors à un calcul de dimensions d'objets de cette catégorie.

AMS 2000 Mathematics subject classification: Primary 11S

Let $X$ be a compact surface such that $Y\hookrightarrow X$ as a separating, strictly pseudoconvex, real hypersurface;

$$ X\setminus Y=X_+\sqcup X_-, $$

where $X_{+}$ ($X_-$) is the strictly pseudoconvex (pseudoconcave) component of the complement. Suppose further that $X_-$ contains a positively embedded, compact curve $Z$. Under cohomological hypotheses on $(X_-,Z)$ we show that if $\dbarb'$ is a sufficiently small, embeddable deformation of the CR-structure on $Y$, then

$$ \mathrm{R-ind}(\bar{\partial}_{b},\bar{\partial}_{b}')\geq-[\dim H^{0,2}(X_-)+\dim H^0(Z,\mathcal{O}_Z)]. $$

This implies that the set of small, embeddable deformations of the CR-structure on $Y$ is closed, in the $\mathcal{C}^{\infty}$-topology on the set of all deformations.

AMS 2000 Mathematics subject classification: Primary 32V15; 32V30. Secondary 32G07; 32W10; 32W05

Segre proved that a smooth cubic surface over $Q$ is unirational if and only if it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

AMS 2000 Mathematics subject classification: Primary 14G05; 14G15. Secondary 11G25; 11D25

The integrable model corresponding to the $\mathcal{N}=2$ supersymmetric SU$(N)$ gauge theory with matter in the symmetric representation is constructed. It is a spin chain model, whose key feature is a new twisted monodromy condition.

AMS 2000 Mathematics subject classification: Primary 53D30; 53D20. Secondary 37J35

Let $G$ be a Lie groupoid over $M$ such that the target-source map from $G$ to $M\times M$ is proper. We show that, if $\mathcal{O}$ is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction $G|_\mathcal{U}$ of $G$ to some neighbourhood $\mathcal{U}$ of $\mathcal{O}$ in $M$ is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid $G|_\mathcal{O}$ on the normal bundle $N\mathcal{O}$. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type.

AMS 2000 Mathematics subject classification: Primary 58H05. Secondary 57R99