Continuity of delta invariants and twisted Kähler–Einstein metrics

We show that delta invariant is a continuous function on the big cone. will also introduce an analytic in terms of optimal exponent Moser–Trudinger inequality and prove it varies continuously Kähler cone, from which we deduce continuity greatest Ricci lower bound. Then building work Berman–Boucksom–Jonsson, obtain uniform Yau–Tian–Donaldson theorem for twisted Kähler–Einstein metrics transcendental cohomology classes.