# The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

@article{Li2019TheUV, title={The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties}, author={Chi Li and Gang Tian and Feng Wang}, journal={Peking Mathematical Journal}, year={2019}, volume={5}, pages={383-426} }

We prove the following result: if a $$\,\,\,\,\,{\mathbb {Q}}\,\,\,\,\,$$ Q -Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean…

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## References

SHOWING 1-10 OF 59 REFERENCES

### On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties

- MathematicsCommunications on Pure and Applied Mathematics
- 2020

We prove the Yau‐Tian‐Donaldson conjecture for any ℚ‐Fano variety that has a log smooth resolution of singularities such that a negative linear combination of exceptional divisors is relatively ample…

### Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2019

We prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is…

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

- MathematicsCommunications on Pure and Applied Mathematics
- 2022

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…

### A variational approach to the Yau–Tian–Donaldson conjecture

- Mathematics
- 2015

We give a new proof of a uniform version of the Yau-Tian-Donaldson conjecture for Fano manifolds with finite automorphism group, and of the semistable case of the conjecture. Our approach does not…

### On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties

- MathematicsDuke Mathematical Journal
- 2019

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-Einstein manifolds inside an appropriate Hilbert scheme. In…

### K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics

- Mathematics
- 2012

It is shown that any, possibly singular, Fano variety X admitting a Kähler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of…

### On the K-stability of Fano varieties and anticanonical divisors

- MathematicsTohoku Mathematical Journal
- 2018

We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of…

### Geodesic Rays and K\"ahler-Ricci Trajectories on Fano Manifolds

- Mathematics
- 2014

Suppose $(X,J,\omega)$ is a Fano manifold and $t \to r_t$ is a diverging Kahler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's…

### Geometry and topology of the space of Kähler metrics on singular varieties

- MathematicsCompositio Mathematica
- 2018

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space…

### Tian's properness conjectures and Finsler geometry of the space of Kahler metrics

- Mathematics
- 2015

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has…