Kähler-Einstein metrics on families of Fano varieties

Given a one-parameter family of & Qopf;-Fano varieties such that the central fiber admits a unique K & auml;hler-Einstein metric, we provide an analytic method to show that the neighboring fiber admits a unique K & auml;hler-Einstein metric. Our results go beyond by establishing uniform a priori estimates on the K & auml;hler-Einstein potentials along fully degenerate families of & Qopf;-Fano varieties. In addition, we show the continuous variation of these K & auml;hler-Einstein currents and establish uniform Moser-Trudinger inequalities and uniform coercivity of the Ding functionals. Central to our article is introducing and studying a notion of convergence for quasi-plurisubharmonic functions within families of normal K & auml;hler varieties. We show that the Monge-Amp & egrave;re energy is upper semi-continuous with respect to this topology, and we establish a Demailly-Koll & aacute;r result for functions with full Monge-Amp & egrave;re mass.