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

Let $(X, D)$ be a log variety with an effective holomorphic torus action, and $\Theta$ closed positive $(1,1)$-current. For any smooth function $g$ defined on the moment polytope of we study Monge-Amp\`{e}re equations that correspond to generalized twisted K\"{a}hler-Ricci $g$-solitons. We prove version Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing existence solutions is always equivalent equivariantly uniform $\Theta$-twisted $g$-Ding-stability. When current associated invariant linear system, further show equivariant special test configurations suffice testing stability. Our results allow arbitrary klt singularities generalize most previous (uniform) YTD (twisted) K\"{a}hler-Ricci/Mabuchi solitons or K\"{a}hler-Einstein metrics.