Abstract:

The Grothendieck comparison theorem gives rise to a de Rham--Betti structure on the cohomology of a smooth projective variety over \bar{\Q}. This interaction between the algebraic de Rham cohomology and the Betti cohomology is encoded in the periods. We follow the work of Andre, Bost and Charles, to formulate the theory via the category of de Rham--Betti objects. We then discuss its relation with algebraic cycles and state the de Rham--Betti conjecture and the Grothendieck period conjecture. Some known results in abelian varieties and HyperKahler varieties will also be discussed. This is a joint work with T. Kreutz and C. Vial.