Joint Seminar Dijon-Freiburg-Strasbourg
We consider complete intersections inside a variety with finite-dimensional motive for which the Lefschetz standard conjecture B holds. We show how conditions on the (modified) niveau filtration on homology influence the Chow groups. This leads to a generalization of results of Voisin and Vial on injectivity of cycle class and Abel-Jacobi maps. Using a variant involving group actions, we obtain several new examples of complete intersections with finite-dimensional motive. This is joint work with Robert Laterveer and Chris Peters.
The generalized Franchetta conjecture as formulated by O’Grady is about algebraic cycles on the universal K3 surface. It is natural to consider a similar conjecture for algebraic cycles on universal families of hyperkaehler varieties. This has close ties to Beauville’s conjectural ``splitting property’’, and the Beauville-Voisin conjecture (stating that the Chow ring of a hyperkaehler variety has a certain subring injecting into cohomology). I will attempt to give an overview of these conjectures, and present some cases where they can be proven. This is joint work with Lie Fu, Mingmin Shen and Charles Vial.
Grothendieck, Verdier, and Deligne in the 60's observed that classical duality theorems like Poincaré, or Serre duality can be most elegantly expressed, and vastly generalized, by a formalism of the six functors. This makes essential use of derived categories. The latter are, however, not sufficient for the purpose of descent. Descent is essential to define equivariant (co)homology and for equivariant duality theorems, and more generally to extend six-functor-formalisms to stacks, which is very important in applications. The problem with (co)homological descent is that the ``glueing data'' has a higher-categorical nature. In this talk we explain how our theory of fibered derivators, based on the idea of derivator due to Grothendieck and Heller (will be explained as well !), solves the problem of (higher-categorical) descent in a way closely related to the classical theory of cohomological descent (due to Deligne in SGA4). However, it is, in contrast, completely self-dual, making it very suitable for the descent of six-functor-formalisms.
The building is No 2 on this map
The building is No 13 on the above map