arXiv research

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

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

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

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

Counterexample disproves conjectures about log canonical thresholds.

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

Effective Yau-Tian-Donaldson conjecture for spherical varieties.

In [Cheeger-Tian 2005], Cheeger-Tian proved an $ε$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional manifolds and higher dim…

Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.

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 been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the origin…

Research confirms Streets-Tian conjecture for 2-step solvmanifolds.

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

In this paper, we study the behavior of Bergman kernels along the Kähler Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler Ricci flow under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture …

We study Tian's $α$-invariant in comparison with the $α_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section $H$. A conjecture of Tian asserts that $α(S_d,H)=α_1(S_d,H)$. We show that this is indeed true for $d=4$ (the res…

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

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…

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

In this paper, we compute Tian's $α_{m,k}^{K\times K}$-invariant on a polarized $G$-group compactification, where $K$ denotes a maximal compact subgroup of a connected complex reductive group $G$. We prove that Tian's conjecture (see Conjecture 1.1 below) is true for $α_{m,k}^{K\times K}$-invariant on such manifolds wh…

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

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

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

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-…

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the …

Solves Tian's stabilization problem for toric Fano manifolds.

We provide here a counter-example to the second inequality of Corollary (19.10) in the Clay Institute Monograph by J.Morgan and G.Tian entitled "Ricci Flow and the Poincare Conjecture". We had announced the existence of this counter-example in our paper "Five Gaps in Mathematics", Advanced Non-linear Studies, vol 15, N…

The Streets-Tian conjecture is confirmed for specific types of Hermitian manifolds.

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

Uniform Laplace comparison for Kähler Ricci flow on Fano manifolds.

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.

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…

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.

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

Study convexity of Mabuchi functional in big cohomology classes.

In this paper, we apply the Tian-Yau-Zelditch expansion of the Bergman kernel on polarized Kähler metrics to approximate plurisubharmonic functions and compute the $α$-invariant of $CP^2#2\bar{CP^2}$, which is exactly 1/3. In addition we prove Tian's conjecture on the generalized Moser-Trudinger inequality in a special…

This manuscript served as lecture notes for a mini-course in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kahler geometry by describing the recent resolution of Tian's three influential properness conjectures in joint work with T. Darvas. These resul…

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.

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…

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.

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…

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

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…

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

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the su…

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 …

Study extends continuity equation for Gauduchon metrics.

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