Une propriété de continuité associée aux classes de cohomologie de Hodge

The obstructions, for a closed smooth differential form of bidimension (p,p) on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are calculated 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 that are completely determined. Each of these currents is ddc-closed and with support in the intersection of the manifold and of the projective subspace associated with the parameter. By the theory of harmonic forms, a period is thus associated with that differential form for each parameter. We study the set of periods, obtained when the parameter varies, and we arrive at a continuity on the Grassmannian, when the cohomology class is rational. The same property can be obtained by going to the space of divisors of the Grassmannian and by using a characterization of Chow forms. We proceed here directly, by calculating the periods by means of the Atiyah-Hirzebruch theorem. This global continuity implies the orthogonality for all parameter.