Kähler-Einstein metrics on families of Fano varieties

Abstract: Given a one-parameter family of $\mathbb{Q}$-Fano varieties such that the central fibre admits a unique K\"ahler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique K\"ahler-Einstein metric. Our results go beyond by establishing uniform a priori estimates on the K\"ahler-Einstein potentials along fully degenerate families of $\mathbb{Q}$-Fano varieties. In addition, we show the continuous variation of these K\"ahler-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\"ahler varieties. We show that the Monge-Amp`ere energy is upper semi-continuous with respect to this topology, and we establish a Demailly-Koll\'ar result for functions with full Monge-Amp`ere mass.