Pfaff Systems, currents and hulls

TL;DR

This paper proves the existence of positive ddbar-closed currents directed by Pfaff systems on compact complex manifolds without integrability assumptions, and explores related harmonicity, maximum principles, and extension problems.

Contribution

It establishes the existence of directed currents in Pfaff systems without integrability and analyzes their properties using ddbar-negative currents.

Findings

Existence of positive ddbar-closed currents directed by Pfaff systems.

Local singular solutions always exist.

Application to harmonicity and extension problems of currents.

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.