Malgré les succès de la théorie de Hodge non abélienne de Corlette-Simpson pour exclure que de nombreux groupes de présentation finie soient groupes fondamentaux de variétés projectives lisses (ou des groupes Kähleriens), les techniques de construction manquent. La construction de Campana du groupe fondamental orbifold d'une paire orbifolde permet de considérer le groupe fondamental des compactifications orbifolds d'une variété (ou champ) quasiprojective lisse donnée $U$ qui, si quelques précautions sont prises et sous des hypothèses raisonnables - mais pas toujours faciles a vérifier, est un groupe Kählerien. En choisissant bien la variété $U$, les groupes obtenus sont potentiellement intéressants et on utilise souvent des techniques inattendues pour établir les propriétés de leurs représentations linéaires. L'exposé fera un survey de cas particulièrement intrigants ou, par exemple, $U$ est un complément d'arrangement de droites, une variété localement complexe hyperbolique non compacte ou un espace de modules de courbes pointées.[-]

Any finite-dimensional p-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen-Brinon. We generalize their construction to the fundamental group of a $p$-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. When the representation comes from a $Q_{p}$-representation of a $p$-adic Lie group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen-Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan.[-]

Given a $p$-adic local system $L$ on a smooth algebraic variety $X$ over a finite extension $K$ of $Q_{p}$, it is always possible to find a de Rham local system $M$ on $X$ such that the underlying local system $\left.L\right|_{X_{\bar{K}}}$ embeds into $\left.M\right|_{X_{\bar{K}}}$. I will outline the proof that relies on the $p$-adic Riemann-Hilbert correspondence of Diao-Lan-Liu-Zhu. As a consequence, the action of the Galois group $G_{K}$ on the pro-algebraic completion of the étale fundamental group of $X_{\bar{K}}$ is de Rham, in the sense that every finite-dimensional subrepresentation of the ring of regular functions on that group scheme is de Rham. This implies that every finite-dimensional subrepresentation of the ring of regular functions on the pro-algebraic completion of the geometric pi $i_{1}$ of a smooth variety over a number field satisfies the assumptions of the Fontaine-Mazur conjecture. Complementing this result, I will sketch a proof of the fact that every semi-simple representation of $G a l(\bar{Q} / Q)$ arising from geometry is a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.[-]