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
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48 results for “Donaldson metrics”
Link slope stability to Donaldson's functional via Quot-scheme limits.
problem Establishing a connection between slope stability and Donaldson's functional for vector bundles.
method Defining Quot-scheme limits of Fubini-Study metrics and proving Donaldson's functional's coercivity.
result Donaldson's functional is coercive on Fubini-Study metrics for slope stable bundles.
The Donaldson metric is a metric on the space of symplectic two-forms in a fixed cohomology class. It was introduced in [2]. We compute the associated Levi-Civita connection, describe it's geodesics and compute the formula for the covariant Hessian of an energy functional on the space of symplectic structures in a fixe…
Kähler-Einstein metricsmath.DG
Surveying Kähler-Einstein metrics and related conjectures.
problem Understanding Kähler-Einstein metrics and their conjectures.
method Survey and review of existing theories.
result Exploration of the Yau-Tian-Donaldson conjecture for Fano manifolds.
In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space of strongly pseudo-convex complex Finsler metrics on -- a holomorphic vector bundle over a closed Kähler manifold . This Donaldson type functional is a generalization in the complex…
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.
Quantizes symplectic fibrations to analyze vector bundles and metrics.
problem Quantizing higher rank vector bundles and understanding their metrics.
method Relates Berezin-Toeplitz quantization to hybrid systems and symplectic fibrations.
result Established refined estimates for computing balanced metrics on Kähler manifolds.
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.
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.
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.
Proves Yau-Tian-Donaldson conjecture for certain singular Fano varieties.
problem Proving the conjecture for a specific class of singular Fano varieties.
method Using log smooth resolutions and K-polystability criteria.
result Kähler-Einstein metrics exist for K-polystable singular Fano varieties.
New machine learning approach finds Kähler metrics through Grassmannian learning.
problem Finding numerical Kähler metrics through machine learning.
method Gradient descent on the Grassmannian manifold, combining with Donaldson's algorithm.
result Observation of nontrivial local minima in moduli space.
Surveying recent developments in K-stability and metric geometry.
problem Relating constant scalar curvature Kähler metrics to K-stability.
method Use of pluripotential theory and non-Archimedean geometry.
result Interpretation of K-stability in terms of non-Archimedean geometry.
Hermitian-Einstein metrics linked to stability of bundles on orbifolds.
problem Existence of Hermitian-Einstein metrics on stable vector bundles over compact Kähler orbifolds.
method Equivalence of slope stability to the existence of Hermitian-Einstein metrics and properness of a functional.
result Equivalence of Hermitian-Einstein metrics and slope stability for stable vector bundles.
New proof of Donaldson-Uhlenbeck-Yau theorem using variational approach.
problem Proving Donaldson-Uhlenbeck-Yau theorem for slope stable holomorphic vector bundles.
method Variational approach, focusing on Bergman kernel asymptotics.
result Elementary proof of Donaldson-Uhlenbeck-Yau theorem with uniform coercivity.
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…
Machine learning speeds up finding Calabi-Yau metrics.
problem Finding numerical Calabi-Yau metrics efficiently.
method Combining curve fitting and machine learning to approximate Ricci-flat metrics.
result Machine learning can predict Calabi-Yau metrics with minimal training data.
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.
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.
Extends Cheeger-Colding Theory to conic Kahler-Einstein metrics.
problem Proving Yau-Tian-Donaldson conjecture for Fano varieties with specific singularities.
method Extension of Cheeger-Colding Theory.
result Technical tool for proving Y-T-D conjecture.
Study of Ricci-flat metrics on C^2 with cone singularities.
problem Ricci-flat metrics with cone singularities on C^2.
method Gibbons-Hawking ansatz over wedges in R^3.
result Asymptotic behavior and energy computation of constructed metrics.
New approach to proving Chen-Donaldson-Sun theorem with examples.
problem Proving Chen-Donaldson-Sun theorem for families of curves.
method Construction of a special metric on stable vector bundles over surfaces formed by families of curves.
result Demonstrates existence of a special metric related to one-dimensional cycles in moduli space.
In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…
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.
Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stabilitymath.DG
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Equivalence to properness of log -energy and geodesic stability.
result Extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture to cscK cone metrics.
The Space of Kaehler metricsmath.DG
Donaldson conjectured \cite{Dona96} that the space of Kähler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and verify his first part partially. We also prove that the constant scalar curvature me…
In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of …
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature. In [Don02] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson's results and computations…
Gradient estimate proved for Donaldson's equation on Kähler manifolds.
problem Proving gradient estimates for Donaldson's equation on compact Kähler manifolds.
method Using uniform upper bounds for and Alexandrov-Bakelman-Pucci (ABP) maximum principle.
result Gradient estimate for Donaldson's equation derived from uniform bounds.
This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.
problem Describing geometric structures on spaces of stability conditions.
method Introducing Joyce structures and showing their relation to complex hyperkähler structures.
result A Joyce structure on a complex manifold defines a complex hyperkähler structure on its tangent bundle.
The study proves Kähler manifolds with extremal Kähler metrics are relatively K-stable.
problem Existence of extremal Kähler metrics on Kähler manifolds.
method Introducing relative K-stability and proving it for Kähler manifolds with extremal Kähler metrics.
result Kähler manifolds admitting extremal Kähler metrics are relatively K-stable.
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.
We show that if a Fano manifold is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then 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…
Researchers found unique scalar-flat Kähler metrics on toric surfaces.
problem Identifying all scalar-flat Kähler metrics on non-compact toric surfaces.
method Using Donaldson's rephrasing of Joyce's construction, they showed uniqueness.
result The constructed metrics are the only J-complete scalar-flat Kähler metrics on strictly unbounded toric surfaces.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
problem Finding anticanonically balanced metrics on Fano manifolds.
method Simplification of Donaldson's proof using Berezin-Toeplitz quantization.
result Sequence of anticanonically balanced metrics converges to Kähler-Einstein metric.
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.
Recently, Donaldson proved asymptotic stability for a polarized algebraic manifold with polarization class admitting a Kähler metric of constant scalar curvature, essentially when the linear algebraic part of is semisimple. The purpose of this paper is to give a generalization of Donaldson's result t…
In this note, we consider a sequence of test configurations compatible with a Kaehler metric in on a polarized algebraic manifold . Then an explicit formula for the Donaldson-Futaki invariant for the sequence will be given.
Paper proves conditions for existence of constant scalar curvature Kähler metrics.
problem Existence of constant scalar curvature Kähler metrics under specific automorphism conditions.
method Derives estimates for scalar curvature type equations and applies to prove conjectures.
result Proves Donaldson's conjecture on geodesic stability and existence of cscK.
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.
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.
The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric.…
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.
On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flowsmath.DG
Sharp bounds for Calabi energy derived from geodesic rays and Mabuchi K-energy.
problem Sharp lower bounds for Calabi energy in Kähler geometry.
method Using geodesic rays and Mabuchi K-energy, derived a sharp bound for Calabi energy.
result Sharp bound for Calabi energy derived from geodesic rays and Mabuchi K-energy is proven to be sharp.
New insights into symplectic singularities via canonical torus actions.
problem Understanding symplectic singularities and their actions.
method Use of actions and Donaldson-Sun theory.
result Symplectic singularities admit canonical actions.
Defines geodesic-Einstein metrics and their relation to nonlinear stabilities.
problem Nonlinear stabilities of line bundles over holomorphic fibrations.
method Introduces geodesic-Einstein metrics and a Donaldson type functional.
result Geodesic-Einstein metrics minimize the Donaldson type functional.
Existence of Ricci flat metric on Kummer K3 surface proven.
problem Proving existence of Ricci flat metric on Kummer K3 surface.
method General strategy of Donaldson's gluing construction, compact elliptic theory on usual Hölder and Sobolev spaces, explicit isometry to Gibbons-Hawking ansatz.
result Existence of a Ricci flat metric on the Kummer K3 surface.