We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
"Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds." Tohoku Math. J. (2) 71 (4) 525 - 532, 2019. https://doi.org/10.2748/tmj/1576724791