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Duality of positive currents and plurisubharmonic functions in calibrated geometry
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 5, October 2009
- pp. 1211-1239
- 10.1353/ajm.0.0074
- Article
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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.