arXiv research

The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.

Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.

Effective Yau-Tian-Donaldson conjecture for spherical varieties.

Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.

We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-…

Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.

Authors discuss complex and non-Archimedean geometry, proving a conjecture.

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian the…

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

We prove the Yau-Tian-Donaldson's conjecture for any $\mathbb{Q}$-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…

Study convexity of Mabuchi functional in big cohomology classes.

We survey the theory of Kähler-Einstein metrics, with particular focus on the circle of ideas surrounding the Yau-Tian-Donaldson conjecture for Fano manifolds.

Research connects probabilistic and variational approaches to Kahler-Einstein metrics.

In this note, using the recent compactness results of Tian and Chen-Donaldson-Sun, we prove the K-semistable version of Yau-Tian-Donaldson correspondence for Fano manifolds.

In this paper, assuming that a polarized algebraic manifold $(X,L)$ is strongly K-stable, we shall show that the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.

We show that a polarized affine variety admits a Ricci flat Kähler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many f…

Geodesic rays prove key aspects of cscK metrics existence and stability.

Blowups of Kähler manifolds with extremal metrics inherit such metrics under stability conditions.

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates thi…

Proves finitely generated associated graded rings for valuations on log Fano pairs.

In this paper is to extend the Cheeger-Colding Theory to the class of conic Kahler-Einstein metrics. This extension provides a technical tool for [LTW] in which we prove a version of the Yau-Tian-Donaldson conjecture for Fano varieties with certain singularity.

Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.

This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then tu…

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to ob…

Paper proves Hamilton-Tian conjecture using partial C0-estimate.

We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjectu…

Proves constant scalar curvature Kähler metrics are very general.

In this paper, we will establish a regularity theory for the Kähler-Ricci flow on Fano $n$-manifolds with Ricci curvature bounded in $L^p$-norm for some $p > n$. Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau-Tian-Donaldson …

We prove the following result: if a $\mathbb{Q}$-Fano variety is uniformly K-stable, then it admits a Kähler-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 prev…

Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.

Solves modified conjecture for Fano manifolds using Ding stability.

We extend the Faltings modular heights of abelian varieties to general arithmetic varieties and show direct relations with the Kahler-Einstein geometry, the Minimal Model Program, heights of Bost and Zhang, and give some applications. Along the way, we propose arithmetic Yau-Tian-Donaldson conjecture, an equivalence of…

Proves results on K-stability using arcs and Mabuchi functional.

In this paper, improving a preceding work, we obtain asymptotic polybalanced kernels associated to extremal Kaehler metrics on polarized algebraic manifolds. As a corollary, we have a stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally, related to the Yau-Tian-Don…

Introduces Poisson K-stability for Kähler manifolds and proves existence of constant scalar curvature structures.

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…

In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic automorphisms of $X$. Then the polarization class $c_1(L)$ will be shown to admit an e…

Proves existence of Kähler-Einstein metrics in big cohomology classes.

We establish a new partial $C^{0}$-estimate along a continuity path mixed with conic singularities along a simple normal crossing divisor and a positive twisted $(1,1)$-form on Fano manifolds. As an application, this estimate enables us to show the reductivity of the automorphism group of the limit space, which leads t…

Continuity of delta invariant leads to uniform Kähler-Einstein metrics.

Researchers introduce new energies to study constant scalar curvature metrics.

Established a correspondence for toric fibrations using Delzant polytopes.

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds p…

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general $L^p$ lower bound on the Calabi functional involving…