Extension de courants fermés à l'espace projectif

Abstract

A closed smooth differential (q,q)-form θ in a complex projective manifold X embedded in the projective space Pn gives rise to an “extension” to the projective space as a closed current. This means that the intersection number of this current with the direct image by the canonical injection of X of a smooth test form on X is equal to the value of θ on this test form, and that its restriction to X is obtained by means of the blow up with center X. We define in this way an integral transform given by a kernel which is a closed current in Pn×X, and which is used to bring an answer to a problem of Grothendieck. This transform is still defined for an algebraic cycle of codimension q instead of the differential (q,q)-form. We obtain a characterization of the complete intersections by the positivity of the transform. Moreover, we obtain a version of the Lefschetz theorem on the hyperplane section stating that the cohomology class arises from the hyperplane section when the intersection with X is transverse outside the hyperplane section.