On the Hodge and Bloch Generalized Conjecture

Claire Voisin (Institut de Mathématiques de Jussieu)

I plan to speak on various notions of coniveau (Hodge, geometric …) the strongest one being given by the decomposition of the diagonal in the Bloch-Srinivas-Paranjape style. Then I will discuss the relationship between the generalized Hodge conjecture and the generalized Bloch conjecture (the latter predicting vanishing of Chow groups in a certain range assuming the vanishing of certain Hodge numbers). The main new result is: ‘For very general complete intersections, GHC implies generalized Bloch’. The other direction is of course via the decomposition of the diagonal and goes back to the 80’s.