Preliminaries.- Kahler-Einstein metrics and extremal Kahler metrics.- The character f and its generalization to Kahlerian invariants.- The character f as an obstruction.- The character f as a classical invariant.- Lifting f to a group character.- The character f as a moment map.- Aubin's approach and related results.

Kähler-Einstein metrics and integral invariants

Let $X$ be any $\mathbb{Q}$-Fano variety and ${\rm Aut}(X)_0$ be the identity component of the automorphism group of $X$. Let $\mathbb{G}$ denote a connected reductive subgroup of ${\rm Aut}(X)_0$. We prove that if $X$ is $\mathbb{G}$-uniformly K-stable, then it admits a Kahler-Einstein metric. The converse of this result holds true if $\mathbb{G}$ contains a maximal torus of ${\rm Aut}(X)_0$. These results give generalized versions of Yau-Tian-Donaldson conjecture for arbitrary (singular) Fano varieties. A key new ingredient is a valuative criterion for the $\mathbb{G}$-uniform K-stability.

G-uniform stability and K\"{a}hler-Einstein metrics on Fano varieties

We prove the following result: if a $\mathbb{Q}$-Fano variety is uniformly K-stable, then it admits a Kahler-Einstein metric. We achieve this by modifying Berman-Boucksom-Jonsson's strategy with appropriate perturbative arguments and non-Archimedean estimates. The idea of using the perturbation is motivated by our previous paper.

The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Szekelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.

Optimal destabilization of K-unstable Fano varieties via stability thresholds

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the behind algebraic stability: relative Ding stability. A new arising feature is the case of non-uniformly stable, there is a partial coercivity of the modified Ding functionals, and we obtain singular Mabuchi solitons via variational approach. In the unstable case, we determine the maximal destabilizer which is a simple convex function over the moment polytope, and establish a Moment-Weight equality connecting Calabi-type energy and Berman-Ding invariant.

Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties

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-Ampere equations that correspond to generalized and twisted Kahler-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) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics.

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

We prove that for any $\mathbb{Q}$-Fano variety $X$, the special $\mathbb{R}$-test configuration that minimizes the $H$-functional is unique and has a K-semistable $\mathbb{Q}$-Fano central fibre $(W, \xi)$. Moreover there is a unique K-polystable degeneration of $(W, \xi)$. As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for Kahler-Ricci flow limits on Fano manifolds, which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial Kahler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals of real valuations, which are analogous to the minimization problem for normalized volumes.

Jiyuan Han

Papers

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

Algebraic uniqueness of K\"{a}hler-Ricci flow limits and optimal degenerations of Fano varieties