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-Ampère type. We also discuss extension problems of positive directed currents.