Courants résidus et opérateurs de Monge–Ampère

Abstract

We extend to the case when the support is a possibly singular analytic subvariety the Federer theorem on the structure of the residue current. On another hand, we determine the general law of transformation of the distribution γf associated with a holomorphic map f=(f1,…,fN). In such a way, we arrive at the cohomological interpretation of the fundamental class of an effective analytic cycle, which is not necessarily a local complete intersection. By this same law, we obtain a characterization of the pure dimensional algebraic subsets of Cn, which are complete intersections. We also characterize the complete intersections of codimension q in Pn in terms of the solutions of the singular Monge–Ampère equation in Pq. Lastly, we express the condition on the dimension of the poles of the plurisubharmonic function u, so that the Monge–Ampère operator Q∧dcu has measure coefficients, for all closed positive current Q of bidimension (k,k).