Reduction of the Hodge conjecture to a continuity

作者：Meo Michel^*

来源：Comptes Rendus Mathematique, 2010, 348(11-12): 625-628.

DOI：10.1016/j.crma.2010.03.016

The obstructions, for a closed differential form on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are computed in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents which are completely determined. A parameter does not yield any obstruction if the associated projective subspace meets properly the manifold. The embedding of the manifold is degenerated, in view of applying the characterization of currents associated to algebraic cycles by the Chow transformation. We study the set of periods obtained when the parameter varies, in particular, we prove a continuity result, thanks to the constructibility of the Bernstein polynomial. When the cohomology class is rational, we conjecture that this set is connected.

• 出版日期2010-6