2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/158636This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-foldSection 3.3 has been changed to correct a mistake in the original versionDifferential GeometryKahler geometry on toric manifolds, and some other manifolds with large symmetrytext