Coniveau $2$ complete intersection and effective cones

Claire Voisin (Paris VII)

Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ shows that it has coniveau greater than or equal to $r$ once $n+1>dr$.

The generalized Hodge conjecture predicts that in this case the cohomology of $X$ is supported on a closed algebraic subset of codimension at least $r$. This is essentially unknown for $r$ at least $2$.

In the case where $r=2$, we exhibit a geometric phenomenon in the variety of lines of $X$ explaining the estimate above for coniveau $2$, and show that the generalized Hodge conjecture would be a consequence of the following conjecture on the effective cone of

cycles:

Conjecture: Very mobile subvarieties have their class in the interior of the effective cone.