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.
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48 results for “Yau-Tian-Donaldson correspondence”
The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.
problem Uniform Yau-Tian-Donaldson conjecture for polarized toric manifolds.
method Combinatorial sufficient condition for relative K-polystability.
result Uniform relative K-polystability condition established.
Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.
problem Uniform K-stability and existence of cscK metrics.
method Special Fujita approximations and regularization of entropy functional.
result Uniformly K-stable polarized smooth projective varieties admit cscK metrics.
Established a correspondence for toric fibrations using Delzant polytopes.
problem Existence of extremal Kähler metrics on toric fibrations.
method Using weighted constant scalar curvature Kähler metrics and uniform K-stability.
result Equivalence between extremal metrics and weighted uniform K-stability of Delzant polytopes.
Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence.
result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
Fano varieties get Mabuchi solitons if they have extremal Kähler metrics.
problem Characterizing Fano varieties with Mabuchi solitons.
method Using Yau-Tian-Donaldson type correspondence for v-solitons.
result Existence of Mabuchi solitons linked to existence of extremal Kähler metrics.
The paper establishes a correspondence for projective bundles over curves using test configurations and extremal metrics.
problem Establishing a correspondence for projective bundles over curves using test configurations and extremal metrics.
method Constructing compatible test configurations and using the generalized Calabi ansatz.
result The relative uniform stability of \( (\mathbb{P}(E),[ω]) \) implies the existence of an extremal metric.
Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
problem Proving Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
method Analyzing Monge-Ampère equations corresponding to generalized and twisted Kähler-Ricci g-solitons, proving stability conditions.
result Existence of solutions is equivalent to equivariantly uniform Θ-twisted g-Ding-stability.
Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
problem Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
method Filtration approach to prove the conjecture.
result Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.
Effective Yau-Tian-Donaldson conjecture for spherical varieties.
problem Finding effective K-stability criteria for spherical varieties.
method Formulated an effective variant of the Yau-Tian-Donaldson conjecture and reviewed effective K-stability criteria for spherical varieties.
result Effective K-stability criteria can be computed given combinatorial data.
Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.
problem Proves Yau-Tian-Donaldson conjecture for a specific class of manifolds.
method Uses holomorphic actions of compact Lie groups and combinatorial conditions.
result Equivalence of K-uniform stability and K-stability for spherical varieties.
Establishes Yau-Tian-Donaldson conjecture for weighted metrics.
problem Constant scalar curvature Kähler metrics on polarized projective manifolds.
method Extends Chi Li's work to weighted case, uses a priori estimates and slope formulas.
result Proves Yau-Tian-Donaldson conjecture for weighted extremal Kähler metrics.
Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.
problem Existence of balanced metrics and Gieseker stability of vector bundles.
method Quot-scheme limit of Fubini-Study metrics and Bergman 1-parameter subgroups.
result Existence of balanced metrics equivalent to Gieseker stability for singular cases.
Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.
problem Existence of Kähler-Einstein metrics with arbitrary polarizations.
method Quantization techniques and pluripotential theory.
result Uniform Yau-Tian-Donaldson theorem for Kähler-Einstein metrics.
New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.
problem Donaldson-Uhlenbeck-Yau theorem implications
method Geodesic rays of Hermitian metrics
result New proofs of the theorem
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-…
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
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…
Proves existence of Kähler-Einstein metrics in big cohomology classes.
problem Existence of Kähler-Einstein metrics in big cohomology classes.
method Using a divisorial stability condition and Fujita-Odaka type delta invariants, building up from scratch the theory of pluripotential theory.
result Uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics in the big cohomology class setting.
Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.
problem Complex Monge-Ampère measures and their applications in algebraic geometry.
method Derives formulas and reduces conjectures to simpler existence problems.
result Reduces uniform Yau-Tian-Donaldson conjecture to existence of approximate decompositions.
We prove the Yau-Tian-Donaldson's conjecture for any -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…
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
Continuity of delta invariant leads to uniform Kähler-Einstein metrics.
problem Continuity of delta invariant in Kähler and twisted Kähler-Einstein metrics.
method Analytic delta invariant and uniform Yau-Tian-Donaldson theorem.
result Uniform Yau-Tian-Donaldson theorem for twisted Kähler-Einstein metrics.
Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
problem Exploring connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
method Surveying and reviewing recent works, transforming complex Monge-Ampère equations, and proving the YTD conjecture.
result Established a transformation from irregular Sasaki-Einstein metrics to -solitons on quasi-regular quotients.
Kähler-Einstein metricsmath.DG
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.
Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …
Introduces Poisson K-stability for Kähler manifolds and proves existence of constant scalar curvature structures.
problem Stability conditions for Poisson structures on Kähler manifolds.
method Infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
Emergent complex geometrymath.DG
Research connects probabilistic and variational approaches to Kahler-Einstein metrics.
problem Constructing Kahler-Einstein metrics on complex projective varieties.
method Combines probabilistic construction and variational methods.
result Non-Archimedean geometry of X emerges from probabilistic framework.
In this paper, assuming that a polarized algebraic manifold is strongly K-stable, we shall show that the polarization class admits a constant scalar curvature Kaehler metric.
Blowups of Kähler manifolds with extremal metrics inherit such metrics under stability conditions.
problem Extremal Kähler metrics on blowups of Kähler manifolds.
method Analyzing K-stability and geometric invariant theory.
result Blowups of Kähler manifolds with extremal metrics inherit such metrics under stability conditions.
Finite generation for valuations computing stability thresholds and applications to K-stabilitymath.AG
Proves finitely generated associated graded rings for valuations on log Fano pairs.
problem Stability thresholds of log Fano pairs.
method Proves finite generation of associated graded rings for valuations.
result Log Fano pairs are uniformly K-stable if their stability threshold is less than a certain value.
Paper proves existence of weighted constant scalar curvature metrics.
problem Existence of weighted constant scalar curvature Kähler metrics.
method Coercivity of weighted Mabuchi functional implies existence of wcscK metric.
result Equivalence of coercivity and existence of wcscK metrics.
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.
Geodesic rays prove key aspects of cscK metrics existence and stability.
problem Existence and stability of constant scalar curvature Kähler metrics.
method Reduction to regularization conjecture and analysis of geodesic rays.
result Uniform K-stability and -stability are sufficient for cscK metrics existence.
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
We prove the following result: if a -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…
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…
Extremal Kähler metricsmath.DG
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 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…
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…
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 , we choose a maximal algebraic torus in the group of holomorphic automorphisms of . Then the polarization class will be shown to admit an e…
Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general non-equilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasi-convex N-particle interaction energy. We show that a deterministic "macroscopic" evolution…
In this paper, we will establish a regularity theory for the Kähler-Ricci flow on Fano -manifolds with Ricci curvature bounded in -norm for some . 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 …
Proves results on K-stability using arcs and Mabuchi functional.
problem K-stability in Fano manifolds and uniform K-polystability.
method Arcs and numerical criterion for stability of pairs.
result Characterizes coercivity of Mabuchi functional in terms of K-polystability.
We establish a new partial -estimate along a continuity path mixed with conic singularities along a simple normal crossing divisor and a positive twisted -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…
The paper analyzes Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.
problem Behavior of Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.
method Established Lojasiewicz's type inequality for Perelman's entropy and proved convergence of Kähler-Ricci flow.
result Solved Yau-Tian-Donaldson conjecture and showed the kernel corresponds to local moduli space of modified -semistable Fano manifolds.