Kähler–Einstein metrics on families of Fano varieties

Abstract

Abstract Given a one-parameter family of ℚ-Fano varieties such that the central fiber admits a unique Kähler–Einstein metric, we provide an analytic method to show that the neighboring fiber admits a unique Kähler–Einstein metric. Our results go beyond by establishing uniform a priori estimates on the Kähler–Einstein potentials along fully degenerate families of ℚ-Fano varieties. In addition, we show the continuous variation of these Kähler–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 Kähler varieties. We show that the Monge–Ampère energy is upper semi-continuous with respect to this topology, and we establish a Demailly–Kollár result for functions with full Monge–Ampère mass.