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Recently the authors showed that there is a robust potential theory attachedto 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 paperwe show that, in a strong sense, the plurisubharmonic functions are the polarduals of the \phi-submanifolds, or more generally, the \phi-currents studied inthe original paper on calibrations. In particular, we establish an analogue ofDuval-Sibony Duality which characterizes points in the \phi-convex hull of acompact set K in X in terms of \phi-positive Green's currents on X and Jensenmeasures on K. We also characterize boundaries of \phi-currents entirely interms of \phi-plurisubharmonic functions. Specific calibrations are used asexamples throughout. Analogues of the Hodge Conjecture in calibrated geometryare considered.

Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry