Kähler–Einstein metrics along the smooth continuity method

Abstract

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kahler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kahler–Einstein manifolds. We also give analogous results for twisted Kahler–Einstein metrics and Kahler–Ricci solitons.