Thematic Term Complex Algebraic Geometry

Joseph Ayoub

Titre: Groupes de Galois motiviques relatifs et géométrie rigide

Résumé: Soit $k\subset K$ une extension de corps. Le groupe de Galois motivique de $K$ relativement à $k$, noté $G_{mot}(K/k)$, est le noyau du morphisme de pro-groupes algébriques $G_{mot}(K) \to \G_{mot}(k)$. En utilisant la géométrie rigide, et notamment les motifs des variétés rigides, on montre que $G_{mot}(K)$ est un produit semi-direct de $G_{mot}(K/k)$ par $G_{mot}(k)$ (par exemple si $k$ est algébriquement clos). On donnera aussi un sens à l'énoncé suivant: "G_{mot}(K/k) est un motif sur $k$" et on fera le lien avec la théorie de Hodge non-abélienne.

Jean-Louis Colliot-Thélène

Titre: Cohomologie non ramifiée de degré 3 et cycles de codimension 2

Richard Hain

Title: On the section conjecture for curves over universal function fields.

Abstract: I will discuss a version of Grothendieck's Section Conjecture for the universal curve over the function field of the moduli space of curves type (g,n) with a level m structure, where g>=5, and n>=0, and m >=1. In this variant of the Section Conjecture, the geometric fundamental group of the curve is replaced by its unipotent completion.

Klaus Hulek

Title: Enriques involutions on jacobian elliptic K3 surfaces and applications

Abstract: We shall discuss a geometric construction for Enriques involutions on jacobian elliptic K3 surfaces and relate this to known interesting families of Enriques surfaces. We shall also discuss applications concerning the Brauer group of Enriques surfaces as well as the field of definition of singular Enriques surfaces. This is joint work with Matthias Schütt.

Bruno Kahn

Titre: Principe de Yoneda et décomposition de la diagonale

Stefan Müller-Stach

Title: Hodge classes on families of varieties with trivial canonical bundle".

Abstract: Families of Calabi-Yau varieties or abelian varieties over their moduli spaces sometimes carry interesting algebraic cycles in the generic fibers. In order to detect them one needs to find Hodge classes in cohomology groups of the total space or, via the Leray spectral sequence, in cohomology groups of certain local systems. The purpose of this talk is to explain how to do this in many interesting cases. This is joint work with several people, including del Angel, Sheng, Ye, van Straten and Zuo.

Carlos Simpson

Title: Mixed Hodge theory for the local structure of representation spaces

Abstract: Let $R=R(\pi _1(X),G)$ be the space of representations of $\pi 1(X)$ in a reductive group $G$. If $X$ is smooth complex projective and $\rho$ is the monodromy representation of a variation of Hodge structures, then the formal local ring of $R$ at $\rho$ has a MHS. We explore some possible extensions to higher homotopy.

Jörg Wildeshaus

Title: Interior motive of Hilbert-Blumenthal varieties