On the Yau-Tian-Donaldson conjecture for generalized K\"ahler-Ricci soliton equations

TL;DR

This paper proves a version of the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons, establishing the equivalence between existence of solutions and a specific stability condition in a broad setting.

Contribution

It generalizes the Yau-Tian-Donaldson conjecture to twisted and generalized Kähler-Ricci solitons with arbitrary singularities, linking solution existence to equivariant stability.

Findings

Proves the YTD conjecture for generalized Kähler-Ricci solitons.

Shows stability testing can be done with special test configurations.

Allows for arbitrary klt singularities in the analysis.

Abstract

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