Abstract

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 $K\subset X$ 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.

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