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- Title
Pfaff systems, currents and hulls.
- Authors
Sibony, Nessim
- 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.
- Publication
Mathematische Zeitschrift, 2017, Vol 285, Issue 3/4, p1107
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-016-1740-8