Coniveau 2 Complete Intersections and Effective Cones
Tóm tắt
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 ≥ c once n ≥ dc. The generalized Hodge conjecture (GHC) predicts that the cohomology of X is then supported on a closed algebraic subset of codimension at least c. This is essentially unknown for c ≥ 2. In the case where c = 2, we exhibit a geometric phenomenon in the variety of lines of X explaining the estimate for the coniveau, and show that (GHC) would be implied in this case by the following conjecture on effective cones of cycles of intermediate dimension: Very moving subvarieties have their class in the interior of the effective cone.
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