Claire Voisin, IHES & CNRS, France

Coniveau 2 complete intersections and cones of effective cycles

Abstract

The coniveau of a Hodge structure is the smallest r for which an L^p,r-component of the Hodge decomposition is non trivial. The coniveau of the Hodge structure on cohomology of a projective manifold is supposed to have a geometric interpretation via the generalized Hodge-Grothendieck conjecture. For complete intersections in projective space, the coniveau can be computed usng Griffiths theory of residues. The numerical characterization of coniveau 1 complete intersections is obvious: they are the Fano complete intersections. For coniveau 2, we give here a geometric interpretation of the numerical characterization, involving the geometry of their varietes of lines.

We show that the generalized Hodge-Grothendieck conjecture for them would then be a consequence of knowing that a certain algebraic class on the variety of lines is "big", that is, in the interior of the effective cone.