Demailly-Koll\'ar continuity on klt pairs, and applications to alpha and delta invariants

Abstract

We establish a Demailly-Koll\'ar type continuity theorem for plurisubharmonic functions with respect to adapted measures on normal complex analytic klt pairs. As applications, we prove the equality of the analytic and divisorial versions of the alpha and delta invariants on compact normal K\"ahler klt pairs, thereby completing a program initiated by the second author. We further show that the two local alpha invariants introduced by Guedj and Trusiani for an isolated log terminal singularity coincide and that their common value is the nth root of Li's normalized volume. These identities have geometric consequences. The equality for the delta invariant yields a Yau--Tian--Donaldson type divisorial criterion for the solvability of twisted K\"ahler-Einstein equations in big cohomology classes, without a semipositivity assumption on the twist. The local alpha identity determines algebraically the critical exponent governing the existence of positively curved KE metrics near an isolated log terminal singularity, thus confirming a conjecture of Guedj and Trusiani.