  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.

Differential Geometry;Complex Variables