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

International audienceLet Y be a compact Kähler normal space and α ∈ H1,1BC (Y) a Kähler class. We study metric properties of the space Hα of Kähler metrics in α using Mabuchi geodesics. We extend several results by Calabi, Chen and Darvas previously established when the underlying space is smooth. As an application we analytically characterize the existence of Kähler-Einstein metrics on Q-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties