Geometry and topology of the space of Kahler metrics on singular varieties

Let Y be a compact Kahler normal space and let alpha is an element of H-BC(1,1) (Y) be a Kahler class. We study metric properties of the space H-alpha of Kahler metrics in alpha using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kahler-Einstein metrics on Q-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.