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

Abstract

Letbe a polarized log variety with an effective holomorphic torus action, and be a closed positive torus invariant -current. For any smooth positive function defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci -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 -twisted -Ding-stability. When 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.

Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties

“…The following result was proved by adapting the techniques of MMP from [56; 37; 7]. Theorem 2.46 (see [39]) To test the g-Ding-semistability, or the g-Ding-polystability, of .X; /, it suffices to test over all special test configurations.…”

“…The proof of such formulas depends on a fibration technique in the study of equivariant cohomology. This technique is partly motivated by some construction from our previous work [39], although there are key differences which require more concrete calculations; see Remark 3.2.…”

The existence of the Kähler–Ricci soliton degeneration

“…When this is treated in [HL20, Section 7] using [LX14]. Here, we present a proof that is independent of [LX14].…”

“…To prove the equivalence, observe that if is reduced uniformly Ding stable, then it is K-polystable with respect to by [HL20, Proposition 5.16]. To show it is K-polystable with respect to , first by [HL20b, (168) or (189)], it follows that is K-semistable with respect to .…”

“…Indeed, condition (3) of the theorem holds trivially, since any is of the form for some and, hence, satisfies where . Therefore, admits a Kähler–Ricci soliton by [HL20].…”