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, 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 ⊂ 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.