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 ddbar-closed current T of mass 1 directed by the Pfaff system S. There is no integrability assumption. We also show that local singular solutions exist always. Using ddbar-negative currents, we discuss Jensen measures, local maximum principle and hulls with respect to a cone P of smooth functions in the Euclidean complex space, subharmonic in some directions. The case where 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.