On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations

Abstract

AbstractLet (X,D) be a polarized log variety with an effective holomorphic torus action, and Θ be a closed positive torus invariant (1,1) ‐current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge‐Ampère equations that correspond to generalized and twisted Kähler‐Ricci g‐solitons. We prove a version of the Yau‐Tian‐Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ‐twisted g‐Ding‐stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler‐Ricci/Mabuchi solitons or Kähler‐Einstein metrics. © 2022 Wiley Periodicals, Inc.