Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,\phi). In particular, on X there exist \phi-plurisubharmonic functions, \phi-convex domains, \phi-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 \phi-submanifolds, or more generally, the \phi-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the \phi-convex hull of a compact set K in X in terms of \phi-positive Green's currents on X and Jensen measures on K. We also characterize boundaries of \phi-currents entirely in terms of \phi-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.