The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
problem Conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
method Study of constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler--Einstein manifolds.
result Sufficient conditions for a cscK perturbation of a Kähler--Einstein metric to remain Kähler--Einstein.
New examples found of complex manifolds with special metrics.
problem Existence of Kähler-Einstein metrics on certain complex manifolds.
method Using Hultgren's polytope formulation, constructing explicit examples of toric Fano manifolds.
result Found examples of projective bundles that admit coupled Kähler-Einstein metrics but no ordinary Kähler-Einstein metrics.
Survey on Kähler-Einstein and weighted solitons on Fano manifolds.
problem Existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds.
method Generalization of algebraic conditions for K-polystability.
result Existence of coupled Kähler-Einstein metrics and weighted solitons is equivalent to algebraic conditions.
Paper discusses conditions for deforming coupled Kähler-Einstein metrics.
problem Conditions for deforming coupled Kähler-Einstein metrics.
method Analyzes deformation of coupled Kähler-Einstein metrics on Fano manifolds.
result Necessary and sufficient condition for deformation of coupled Kähler-Einstein metrics.
We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…
Proves existence of Kähler-Einstein metrics with certain twisting forms.
problem Existence of Kähler-Einstein metrics under stability conditions.
method Proves existence assuming twisted K-stability condition, allows certain non-negative twisting forms.
result Existence of twisted Kähler-Einstein metrics under specified conditions.
New bound on partition function proves Kähler-Einstein stability.
problem Proving Kähler-Einstein metrics on complex manifolds.
method Quantitative bound on partition function, connecting probabilistic and quantization approaches.
result Direct analytic proof of Kähler-Einstein stability for uniformly Gibbs stable manifolds.
Extends construction of Kähler-Einstein metrics to noncompact manifolds.
problem Construct complete Kähler-Einstein metrics on noncompact manifolds.
method Iterative construction using Berndtsson's method.
result Induces semipositively curved metric on relative canonical bundle.
The paper examines conditions for Kähler-Einstein metrics on deformations of Fano manifolds.
problem Conditions for Kähler-Einstein metrics on deformations of Fano manifolds.
method Analyzes necessary and sufficient conditions, approximates Weil-Petersson metric, describes plurisubharmonicity of energy functional.
result Provides new conditions for the existence of Kähler-Einstein metrics on deformations of Fano Kähler-Einstein manifolds.
Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
problem Existence and characterization of Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
method Variational approach, algebraic approximation of singularities, function α_ω, continuity method.
result Many K-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities.
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. method Proving existence through compactifications of Lie groups.
result Classification of Q-Fano compactifications of SO4(C) with Kähler-Einstein metrics. Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
Study finds Kähler-Einstein metrics on two Pasquier varieties.
problem Existence of Kähler-Einstein metrics on specific varieties.
method Analyzes Pasquier's two-orbits varieties to find metrics.
result New example of K-unstable Fano manifold with Picard number one.
Study classifies Kähler-Einstein metrics with rotational symmetries.
problem Classify Kähler-Einstein metrics with rotational symmetries.
method Focus on integrable structures to classify metrics.
result Classify Kähler-Einstein metrics with rotational symmetries.
Proves finite step termination of Kähler-Einstein metric singularity formation.
problem Singularity formation of Kähler-Einstein metrics.
method Finite step termination of bubble trees for singularity formation.
result Finite step termination of Kähler-Einstein metric singularity formation proved in non-collapsing situation.
Study two types of singular Kähler-Einstein metrics on complex varieties.
problem Understanding different types of singular Kähler-Einstein metrics.
method Analyzing Ricci flat Kähler cone metrics and applying to more general spaces.
result A weaker notion of singular Kähler-Einstein metrics is equivalent to a stronger one under certain conditions.
Existence of Kähler-Einstein metrics on toric varieties proven.
problem Existence of Kähler-Einstein metrics on toric varieties.
method Characterization of K-stability using log Cox ring and universal orbifold cover.
result Every Q-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric.
We obtain a residue formula for an obstruction to the existence of coupled Kähler-Einstein metrics described by Futaki-Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler-Einstein metric but still admits coupled K…
Researchers compute limits of Kähler-Einstein forms on degenerating manifolds.
problem Understanding limits of Kähler-Einstein forms on degenerating manifolds.
method Hybrid convergence of Kähler-Einstein measures using algebro-geometric limits.
result Limit measure is a weighted sum of Dirac masses at divisorial valuations.
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by h12 and the divergence of the Kodaira-Spencer bracket. New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.
problem Understanding Kähler-Einstein currents on singular spaces.
method Analyzing currents' properties and applying them to compact spaces with log terminal singularities.
result Kähler-Einstein currents dominate Kähler forms on compact spaces with log terminal singularities.
Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.
problem Characterizing Kähler-Einstein metrics induced by projective immersions.
method Analyzing four families of symmetric and non-symmetric toric Fano manifolds.
result Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.
We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being…
In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the va…
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
The paper studies Kähler-Einstein metrics with singularities and their limits.
problem Analyzing Kähler-Einstein metrics with crossing edge singularities and their limits.
method Extending Guenancia's techniques, the paper shows convergence of metrics under specific angle conditions.
result Negatively curved Kähler-Einstein crossing edge metrics converge to mixed cusp and edge metrics smoothly away from the divisor.
The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.
problem Existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. method Analyzes Q-Fano group compactifications, proving uniqueness and existence of Kähler-Einstein metrics. result Proves the existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover,…
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.
problem Verifying a conjecture about Kähler-Einstein edge metrics on Hirzebruch surfaces.
method Using the Calabi ansatz, constructing a family of metrics and studying their angle deformation.
result Verification of a conjecture and finding a rigid singularity.
Constructs Kahler-Einstein metrics near isolated log canonical singularities.
problem Constructing metrics near singularities in complex geometry.
method Constructs Kahler-Einstein metrics with negative scalar curvature near isolated log canonical singularities.
result Metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable.
Solves classical problem with Kähler-Einstein metrics in complex projective spaces.
problem Classical problem of non-isometric bidimensional Kähler-Einstein submanifolds.
method Listed complete non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds.
result Solves the classical problem in the specified case.
Decomposes Q-Fano Kähler-Einstein varieties into simpler components.
problem Understanding the structure of Q-Fano Kähler-Einstein varieties.
method Proves decomposition theorem using algebraically integrable foliations and stability conditions.
result Q-Fano Kähler-Einstein varieties decompose into simpler components.
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.
Study Kähler-Einstein potentials on stable varieties near singularities
problem Asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities
method Using iterated logarithmic functions and refined lower bounds
result Improved estimates for Kähler-Einstein potentials
Establishes a lower bound for Kähler-Einstein distance on certain domains.
problem Finding a lower bound for Kähler-Einstein distance on specific types of domains.
method Proves an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.
result Establishes a lower bound for the Kähler-Einstein distance on pseudoconvex domains with positive hyperconvexity index.
Continuity of Kähler-Einstein potentials at singularities proven.
problem Regularity of solutions to degenerate complex Monge-Ampère equations on singular spaces.
method Investigation of Dirichlet problem and global continuity of solutions.
result Kähler-Einstein potentials are continuous at isolated singularities.
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…
New compact K-E manifolds with negative curvature found.
problem Constructing compact Kähler-Einstein manifolds with negative curvature.
method Created compact Kähler-Einstein manifolds of dimension n with negative sectional curvature.
result Found compact Kähler-Einstein manifolds of negative curvature not covered by the ball.
The paper discusses polynomial convergence to conical Kähler-Einstein metrics.
problem Understanding the convergence of Kähler-Einstein metrics to conical structures.
method Two-step degeneration theory and algebraic singularity analysis.
result Singular Kähler-Einstein metrics are conical if curvature grows quadratically near a point.
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
problem Existence of complete holomorphic vector fields on complex manifolds with specific metrics.
method Method of potential scaling to find a potential function with constant length differential, then constructing a vector field from its gradient.
result A complete holomorphic vector field is constructed on a complex manifold with a Kähler-Einstein metric.
The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.
problem Classifying Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. method Proving finiteness through classification of compactifications.
result There are only finitely many Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.
problem Finding Kähler-Einstein metrics on specific Fano varieties.
method Using combinatorial criteria for K-polystability and properties of Fano varieties.
result Proves existence of Kähler-Einstein metrics on Xm for m≥4 and on Ym for m=4,5. We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.
New proof of Kähler-Einstein Fano manifold L∞ estimates.
problem Uniform L∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds. method Using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle, without pluripotential theory.
result Uniform L∞ estimates derived for Kähler-Ricci flow on Kähler-Einstein Fano manifolds. The unit ball is characterized by a Kähler-Einstein potential.
problem Characterizing the unit ball in complex geometry.
method Using a global potential function of the Kähler-Einstein metric.
result A compact Kähler manifold with an ample canonical bundle is the unit ball if it has a specific potential function.
Study degenerations of Kähler-Einstein metrics on surfaces.
problem Understanding the geometry of Kähler-Einstein metrics on surfaces as they degenerate.
method Construct a Kähler-Einstein neck region to model degeneration.
result Provides a model for the limiting geometry of metrics in the family.