Pfaff systems, currents and hulls

Abstract

Let S be a Pfaff system of dimension 1, on a compact complex manifold M. We prove that there is a positive \({\partial \overline{\partial }}\)-closed current T of bidimension (1, 1) and of mass 1 directed by the Pfaff system S. There is no integrability assumption. We also show that local singular solutions always exist. Under a transversality assumption of S on the boundary of an open set U, we prove the existence in U of positive \({\partial \overline{\partial }}\)-closed currents directed by S in U. Using \(i{\partial \overline{\partial }}\)-negative currents, we discuss Jensen measures, local maximum principle and hulls with respect to a cone \(\mathcal P\) of smooth functions in the Euclidean complex space, subharmonic in some directions. The case where \(\mathcal P\) is the cone of plurisubharmonic functions is classical. We use the results to describe the harmonicity properties of the solutions of equations of homogeneous, Monge-Ampere type. We also discuss extension problems of positive directed currents.