Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry
arXiv:0710.3921
Abstract
Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,ϕ). In particular, on X there exist ϕ-plurisubharmonic functions, ϕ-convex domains, ϕ-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the ϕ-submanifolds, or more generally, the ϕ-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the ϕ-convex hull of a compact set K in X in terms of ϕ-positive Green's currents on X and Jensen measures on K. We also characterize boundaries of ϕ-currents entirely in terms of ϕ-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.
Minor typographical errors have been corrected