Coniveau 2 Complete Intersections and Effective Cones |
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Authors: | Claire Voisin |
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Affiliation: | 1. CNRS and Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013, Paris, France
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Abstract: | Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface X of degree d in mathbbPn{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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