  Motivated by constructions appearing in mirror symmetry, we study special
representatives of complexified K\"ahler classes, which extend the notions of
constant scalar curvature and extremal representatives for usual K\"ahler
classes. In particular, we provide a moment map interpretation, discuss a
possible correspondence with compactified Landau-Ginzburg models, and prove
existence results for such special complexified K\"ahler forms and their large
volume limits in certain toric cases.
