English
FrançaisTitres et résumés
Donu Arapura
Title: Motivic sheaves and direct images
Abstract: I want to outline a construction of an abelian category of motivic sheaves over a characteristic 0 base variety. This is based on a method due to Nori. Since many people have heard versions of this talk, I would like to concentrate on some of the more technical aspects involving construction of direct images.
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.
Francis Brown
Title: Motives and periods of graph hypersurfaces
Abstract: I will give a survey of recent progress and open problems concerning the motives of graph hypersurfaces.
François Charles
Titre: Quelques aspects arithmétiques des fonctions normales
Résumé: Des résultats récents de Brosnan-Pearlstein, Schnell et M. Saito montrent que le lieu des zéros d’une fonction normale est une variété algébrique. On présentera dans cet exposé quelques propriétés arithmétiques de ce lieu en prouvant, dans un cadre variationnel, quelques théorèmes de comparaison entre les noyaux des applications d’Abel-Jacobi complexe et l-adique et en essayant de dégager une notion de fonction normale pour la cohomologie étale.
Jean-Louis Colliot-Thélène
Titre: Cohomologie non ramifiée de degré 3 et cycles de codimension 2
Hélène Esnault
Title: Some aspects of the algebraic fundamental goup in characteristic 0 and $p>0$, and of its relation to its profinite completion
Abstract: The homomorphism $pi^{alg}(X) \to pi^{et}(X)$ for a smooth complex manifold has a $k$ structure if $X$ is defined over a subfield $k$ of the complex numbers. Here $\pi^{all}(X)$ is the algebraic completion of the topological fundamental group. We discuss this in char. $p>0$ from two viewpoints: how the étale fundamental group controls the algebraic one on $X$ projective smooth over an algebraically closed characteristic $p>0$ field (Gieseker conjecture, joint with V. Mehta), and how to extend the algebraic one to recover Nori's fundamental group as the profinite quotient (joint with A. Hogadi).
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
Marc Levine
Title: Motives and generalized cohomology of algebraic varieties
Abstract: Thanks to Morel-Voevodsky, we now have algebraic analogs of classical stable homotopy theory, and with this, algebraic analogs of generalized cohomology. The analog of ordinary cohomology (e.g. singular cohomology) are various versions of motivic cohomology. In classical homotopy theory, the Postnikov tower breaks up a generalized cohomology theory into its ordinary pieces, while in motivic homotopy theory, Voeovodsky's version of the Postnikov tower breaks up a generalized cohomology theory into motives. We describe some of these results, as well as more recent results in motivic ``semi-stable" homotopy theory.
Eduard Looijenga
Title: The KZ system as a variation of mixed Hodge structure
Abstract: We explain how a KZ system can be understood as a variation of mixed Hodge structure. A main tool is an algebro-geometric realization of the irreducible highest weight representations of a simple Lie algebra (or more generally, of one of Kac-Moody type).
Ben Moonen
Title: Special subvarieties in the Torelli locus
Abstract: The problem that I want to review and discuss is whether there are, for a given genus, special subvarieties (a.k.a. subvarieties of Hodge type) that are contained in the Torelli locus. The talk will have the character of an overview. In particular, I'll discuss the nontrivial known examples, and the known restrictions.
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.
Johannes Nicaise
Title: Degenerations of Calabi-Yau varieties and monodromy
Madhav Nori
Title: Tits Buildings and Volodin K-theory
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.
Vasudevan Srinivas
Title: Algebraic cycles on a generic complex abelian 3-fold
Summary: This is a report on joint work with A. Rosenschon. We show that on such a 3-fold, for all but a finite number of positive integers $n$, the CHow group of curves with mod $n$ coefficients is not finitely generated. This is done in two steps: first we use a varaiant of the technique of Bloch and Esnault to show that the Ceresa cycle is not $n$-divisible for almost all $n$. Then we use modular correspondences, following Nori, to show infinite generation.
Burt Totaro
Title: Bounds for the Chow ring of a classifying space
Abstract: The classifying space BG of an algebraic group G is a central example for trying to understand torsion phenomena in A1-homotopy theory. In the 1990s, I gave explicit bounds on the degrees of generators of the Chow ring of BG, for all G. At the time, no bound was known for the degrees of generators of the cohomology ring of a finite group, but Symonds has recently proved essentially the same bound for cohomology.
The lecture will give a better bound for the Chow ring of any finite group. One application is to exhibit a fairly wide class of finite groups whose Chow ring is generated by transfers of Chern classes.
Jörg Wildeshaus
Title: Interior motive of Hilbert-Blumenthal varieties
Abstract: Bondarko has recently introduced the notion of weight structure on a triangulated category. He shows that the category $DM$ of motives à la Voevodsky carries a canonical such structure, and that its heart is identical to the category $CHM$ of Chow motives.
This result allows to show that the inclusion of $CHM$ into $DM$ admits an adjoint functor, the "part of weight zero" as soon as one restricts the image $DM$ to a certain sub-category $DM'$ of motives avoiding certain weights.
In a given situation, the question is thus to show that a motive $M$ belongs to $DM'$. If $M$ occurs in the motive of a smooth variety $X$, we shall identify a criterion (C) on the boundary motive of $X$ ensuring that $M$ is in $DM'$, and that hence its "part of weight zero" is defined.
For Hilbert-Blumenthal varieties $X$, the boundary motive is sufficiently understood to allow for an identification of those cases where criterion (C) is satisfied.
Olivier Wittenberg
Titre: Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque
Résumé: Soit X une variété propre et lisse sur un corps de nombres. Des conjectures prédisant l'existence de zéro-cycles sur X remplissant des conditions locales prescrites furent formulées dans les années 80 et 90 par Kato, Saito, Colliot-Thélène. Grâce aux travaux de Saito, Salberger, Colliot-Thélène, Frossard et van Hamel, ces conjectures sont connues dans le cas où X est fibrée en variétés de Severi-Brauer d'indice sans facteur carré au-dessus d'une courbe de groupe de Tate-Shafarevich fini. Dans cet exposé je présenterai une extension de ces résultats à des fibrations plus générales.
Kang Zuo
Title: Relative proportionality for special subvarieties of moduli spaces of abelian varieties and thickenings of Higgs bundles
Abstract: I shall talk on a joint project with Stephan Mueller-Stach and Eckart Viehweg. The relative proportionality principle of Hirzebruch and Hofer was discovered to characterize Shimura curves in a Picard modular surface. It can be expressed as an equality of the Chern classes of the curve and its normal bundle. A similar equality holds for Shimura curves in Hilbert modular surfaces. We prove a generalization of this result to special subvarieties of Shimura varieties of orthogonal type. Furthermore we study the "inverse problem" of deciding when an arbitrary subvariety is a special subvariety, provided it contains sufficiently many special subvarieties, which satisfy relative proportionality.
