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.
Criterion found for Kähler Einstein metrics on toric Fano manifolds.
problem Existence of Kähler Einstein metrics on toric Fano manifolds.
method Criterion based on uniform stability in GIT and properness of a functional.
result Complete criterion for existence of generalized Kähler Einstein metrics.
Unique conical Kähler-Einstein metric found on Fano manifold.
problem Existence of conical Kähler-Einstein metrics on Fano manifolds.
method Proved existence of a unique conical Kähler-Einstein metric with admissible cone angles.
result Existence and uniqueness of conical Kähler-Einstein metric.
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.
Study of compactifications for Kähler-Einstein Fano manifolds.
problem Compactification of moduli spaces of Kähler-Einstein Fano manifolds.
method Geometry of metric tangent cones and algebro-geometric study of singularities.
result First concrete examples of Gromov-Hausdorff compactifications in complex dimensions >2.
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.
New metrics proposed for Kähler manifolds, proving existence and uniqueness.
problem Finding canonical metrics on Kähler manifolds.
method Introducing coupled Kähler-Einstein metrics and proving existence and uniqueness.
result Existence and uniqueness of coupled Kähler-Einstein metrics in specific cases.
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.
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.
Proves conic version of YTD conjecture on log Fano manifolds.
problem Existence of conic Kahler-Einstein metrics on log Fano manifolds.
method Proof of the YTD conjecture for conic Kahler-Einstein metrics.
result Proven existence of conic Kahler-Einstein metrics on log Fano manifolds.
Proof of flow convergence on Fano manifolds.
problem Convergence of Kahler-Ricci flow on Fano manifolds.
method Recent techniques in geometry and analysis.
result Convergence of Kahler-Ricci flow on Fano manifolds.
Proves Matsushima's theorem for Kähler-Einstein metrics on Fano manifolds with cone singularities.
problem Proving Matsushima's theorem for Kähler-Einstein metrics on Fano manifolds with cone singularities.
method Alternative proof using continuity method and existence theorem via conic Ding functional.
result Existence of Kähler-Einstein cone metrics on Fano manifolds with cone singularities.
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. We construct a geometrically compactified moduli algebraic space of Kahler-Einstein Fano manifolds.
We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …
We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.
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.
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.
Study bounds Kähler-Einstein Fano volumes with singularities.
problem Bounding the volume of Kähler-Einstein Fano varieties with singularities.
method Using local singularity invariants and recent results on K-semistability.
result Sharp volume upper bounds for Kähler-Einstein Fano varieties with quotient singularities.
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.
Proof confirms condition for Kähler-Einstein metrics on toric Fano manifolds.
problem Existence of Kähler-Einstein metrics on toric Fano manifolds.
method Condition in terms of barycenters of polytopes.
result Necessary and sufficient conditions for existence of coupled Kähler-Einstein metrics and soliton solutions.
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.
Study on Fano manifolds without K-E metrics and their properties.
problem Characterizing Fano manifolds without K-E metrics and understanding their properties.
method Examining various examples of horosymmetric manifolds and using different constructions to provide infinite families of Fano manifolds.
result Infinitely many examples of Fano manifolds without K-E metrics but with coupled K-E metrics.
Formula found for Kähler-Einstein metric existence obstruction.
problem Obtaining Kähler-Einstein metrics on manifolds.
method Residue formula for Futaki-Zhang obstruction.
result Found coupled Kähler-Einstein metrics on a specific toric Fano manifold.
Mabuchi's metric correlates with a specific stability condition for Fano manifolds.
problem Characterizing Fano manifolds with Mabuchi's soliton metric.
method Proving Mabuchi's metric corresponds to relative D-stability.
result Fano manifolds admit Mabuchi's metric if and only if they are uniformly relatively D-stable.
Kahler-Einstein metrics linked to eigenvalue gaps on Fano manifolds.
problem Existence of Kahler-Einstein metrics on Fano manifolds.
method Characterization via eigenvalue gaps of Cauchy-Riemann and Hamiltonian vector fields.
result Existence of Kahler-Einstein metrics linked to eigenvalue gaps.
Fano manifolds with specific alpha invariant are K-stable.
problem Characterizing Fano manifolds with a specific alpha invariant.
method Analyzing alpha invariant and proving K-stability.
result Fano manifolds with α(X)=n/(n+1) and n≥2 are K-stable. The paper proves conditions for K-stability and existence of Kähler-Einstein metrics on Fano spherical varieties.
problem Conditions for K-stability and existence of Kähler-Einstein metrics on Fano spherical varieties.
method Criterion based on moment polytope and combinatorial data for K-stability; uses Yau-Tian-Donaldson conjecture for Fano manifolds.
result Criterion for K-stability and existence of Kähler-Einstein metrics on spherical Fano manifolds.
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.
Conditions for solutions to complex Monge-Ampère equations on Fano manifolds.
problem Existence of solutions to complex Monge-Ampère equations on Fano horosymmetric manifolds.
method Necessary and sufficient conditions derived from combinatorial data.
result Conditions for existence of solutions in terms of combinatorial data.
In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit …
The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.
problem Analyzing conical Kähler-Einstein metrics on rank one horosymmetric Fano manifolds.
method Investigates the convergence of these metrics as the cone angle approaches zero, focusing on the behavior on the complement of a codimension one orbit.
result The metrics converge to the Kähler-Einstein metric on the basis and Stenzel's Ricci flat Kähler metrics on the fibers.
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…
Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.
problem Characterizing compact complex manifolds with holomorphic GL(2)-geometry.
method Analyzing Kähler-Einstein and Fano manifolds, using GL(2) and SL(2) geometries.
result Only compact Kähler-Einstein manifolds with holomorphic GL(2)-geometry are covered by compact complex tori, three dimensional quadric, or three dimensional Lie ball.
The paper proves stability of Kähler-Ricci flows on Fano manifolds.
problem Stability of conical Kähler-Ricci flows on Fano manifolds.
method Using the boundedness of Log Mabuchi energy, the paper proves stability of conical Kähler-Einstein metrics.
result For any β' close to β, the conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric.
Proves existence of Kähler-Einstein metrics on complex manifolds.
problem Existence of Kähler-Einstein metrics on complex manifolds with ample canonical bundle.
method Method of continuity, reducing to C^0 estimate.
result Existence of coupled Kähler-Einstein metrics proven.
Study Fano fibrations and Kähler-Einstein metrics on their bases.
problem Understand geometric structures and metrics on Fano fibrations.
method Construct (1,1)-forms and solve twisted Kähler-Einstein equations. result Singular Kähler metrics on Fano fibrations satisfy twisted Kähler-Einstein equations.
In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.
We show that any n-dimensional Fano manifold X admitting Kähler-Einstein metrics satisfies that the anti-canonical volume is less than or equal to the value (n+1)n. Moreover, the equality holds if and only if X is isomorphic to the n-dimensional projective space.
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. Researchers prove existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.
problem Existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.
method Analyzing automorphisms and limits at various scales to prove the existence of metrics.
result Existence of Kähler-Einstein metrics with conic singularities for β>β∗ close to β∗. The inverse Monge-Ampere flow helps find Kahler-Einstein metrics.
problem Finding Kahler-Einstein metrics on complex manifolds.
method Gradient flow of the Ding energy functional on Kahler metrics.
result The flow converges to a Kahler-Einstein metric in various cases.
Study proves most Fano threefolds are Kähler-Einstein.
problem Existence of Kähler-Einstein metrics on smooth Fano threefolds.
method Investigated smooth Fano threefolds of Picard rank one and anticanonical degree 22 with C∗ action. result Proved existence of Kähler-Einstein metrics on most such threefolds, except possibly two cases.
In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space Mˉ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on Mˉ an…
Survey on Kähler-Einstein metrics on smoothable Fano varieties.
problem Existence of Kähler-Einstein metrics on Fano varieties.
method Analyzing properties of Fano varieties and their moduli spaces.
result Construction of compact algebraic moduli spaces of K-polystable Fano varieties.
We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound R(X) to the existence of conical Kahler-Einstein metrics on a Fano manifold X. In particular, if D∈∣−KX∣ is a smooth simple divisor and the Mabuchi K-energy is bounded below, then there exists a unique conical Kahler-Eins…
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. New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.
problem Identifying Fano threefolds of degree 22 with Kähler-Einstein metrics.
method Analyzing the structure and properties of Fano threefolds of Picard rank one with anti-canonical degree 22.
result All remaining Fano threefolds of degree 22 admit Kähler-Einstein metrics.