The metric completion of the Riemannian space of K\"{a}hler metrics

Let X be a compact K ̈ahler manifold and α ∈ H 1 , 1 ( X, R ) a K ̈ahler class. We study the metric completion of the space H α of K ̈ahler metrics in α , when endowed with the Mabuchi L 2 -metric d . Using recent ideas of Darvas, we show that the metric complet ion ( H α , d ) of ( H α , d ) is a CAT(0) space which can be identified with E 2 ( α ), a subset of the class E 1 ( α ) of positive closed currents with finite energy. We further prove, in the toric setting, that H α,tor = E 2 tor ( α ).