Special representatives of complexified Kähler classes

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified Kähler classes, which extend the notions of constant scalar curvature and extremal representatives for usual Kähler 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 Kähler forms and their large volume limits in certain toric cases.