We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a -equivariant Fano compactification of a complex connected reductive group in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …
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48 results for “Kähler-Einstein Fano manifolds”
In this paper, we show that any compact Khler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Khler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold homotopic to a compact Riemannian manifold with negative sectional curva…
The paper classifies Fano distributions on specific Fano manifolds.
problem Investigating Fano distributions on Fano manifolds.
method Classification of Fano distributions on various Fano manifolds.
result Classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric locally away from singular set.
Homogeneous Fano manifolds with specific bundle properties proven.
problem Characterizing Fano manifolds with nef tangent bundles.
method Proving homogeneity for specific Fano manifolds.
result Any Fano manifold of coindex three with nef tangent bundle is homogeneous.
New toric Fano manifolds found without extremal Kähler metrics.
problem Finding toric Fano manifolds without extremal Kähler metrics.
method Constructing specific toric Fano manifolds of dimensions 10 and n (n≥11) that do not admit extremal Kähler metrics.
result Existence of toric Fano manifolds of dimension 10 and higher that do not admit extremal Kähler metrics.
Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.
problem Analyzing the limits of Kähler-Ricci flow on Fano G-manifolds.
method Proves the Gromov-Hausdorff limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
result The limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
Characterizes stable toric Fano manifolds using modified Ding functional.
problem Stability of toric Fano manifolds.
method Characterization through modified Ding functional and pseudo-boundedness analysis.
result Characterization of relative Ding stable toric Fano manifolds.
Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifoldsmath.DG
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 proves Mabuchi solitons and constants on Fano admissible manifolds.
problem Existence of Mabuchi solitons on Fano admissible manifolds.
method Defined Mabuchi solitons and constants, proved existence and non-existence.
result Fano admissible manifolds admit Mabuchi solitons if and only if the Mabuchi constant is less than 1.
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.
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.
The paper studies Kähler-Ricci flow on Fano manifolds and finds examples of type II singularities.
problem Analyzing the behavior of Kähler-Ricci flow on Fano manifolds.
method Proving the flow is of type II and finding specific examples.
result Found examples of Fano compactifications where Kähler-Ricci flow develops type II singularities.
Constructs models for Fano threefolds using Lagrangian torus fibrations.
problem Models for Fano threefolds using Lagrangian torus fibrations.
method Toric degeneration for affine manifolds with singularities, correspondence between polytopes and Fano manifolds.
result Total space of each fibration is homeomorphic to the expected Fano threefold and numerical invariants coincide.
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.
New proof for Fano manifolds, showing rigidity and stability.
problem Proving rigidity and stability of Fano manifolds.
method Birational superrigidity and K-stability approach.
result Projectively normal Fano manifolds of index 1 are birationally superrigid and K-stable.
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.
Quantizes Kähler-Ricci flow for Fano manifolds.
problem Optimal degeneration for Fano manifolds.
method Geometric quantization of Kähler-Ricci flow and entropy functional.
result Established convergence to original flow and entropy.
Compactifies Calabi-Yau to weak Fano manifolds.
problem Compactifying Calabi-Yau manifolds to weak Fano manifolds.
method Generalized Tian-Yau construction and asymptotically Calabi metrics.
result Calabi-Yau structure arises from compactification.
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.
Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitonsmath.DG
Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.
problem Lower boundedness of modified K-energy on Fano manifolds.
method Extend Tosatti's method to study Fano manifolds with Kähler-Ricci solitons.
result Establish lower bounds on modified K-energy for Kähler-Ricci solitons.
Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.
problem Eigenvalue estimates for the Hodge Laplacian on Fano manifolds.
method Application of Bochner-Kodaira formula to study the geometry of Fano manifolds.
result The original Kähler form remains Kähler for deformations, and an explicit Ricci potential is given.
The paper classifies stable toric Fano manifolds and compares -stability and Ding stability.
problem Classifying stable toric Fano manifolds in low dimensions.
method Using Mabuchi constants calculated from moment polytopes and Bott tower structure.
result List of uniform relative Ding stability for toric Fano manifolds up to four dimensions.
Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.
problem Analyzing the behavior of Kähler-Ricci flow on spherical Fano manifolds.
method Gromov-Hausdorff limit and torus degeneration.
result The limit of Kähler-Ricci flow on spherical Fano manifolds is a spherical Fano variety with a Kähler-Ricci soliton.
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.
Paper shows convergence of Fano Kähler-Ricci solitons without uniform Futaki invariant bound.
problem Degeneration of Fano Kähler-Ricci solitons without uniform Futaki invariant bound.
method Improves Phong-Song-Sturm's result by removing the uniform bound assumption.
result Convergence of Fano Kähler-Ricci solitons to a Kähler-Ricci soliton on a Q-Fano variety with log terminal singularities.
We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahl…
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 …
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 confirms Hamilton-Tian conjecture for specific Sasakian manifolds.
problem Hamilton-Tian conjecture for specific Sasakian manifolds.
method Sasaki-Ricci flow, compact transverse Fano Sasakian 5-manifolds, klt foliation singularities.
result Confirmed Hamilton-Tian conjecture for compact transverse Fano Sasakian 5-manifolds.
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.
Eta invariant computed for circle bundles over Fano manifolds.
problem Computing eta invariant for circle bundles over Fano manifolds.
method Using spin-c Dirac operator and adiabatic limit.
result Eta invariant computed for arbitrary adiabatic parameters.
Researchers create a new moduli space for Fano manifolds with special geometric properties.
problem Constructing a new moduli space for Fano manifolds with Kähler-Ricci solitons.
method Developed a moment map picture and used complex analytic charts to construct the moduli space.
result Created a larger moduli space that includes Fano manifolds with Kähler-Einstein metrics.
The paper studies how certain solitons on Fano manifolds extend to nearby deformations.
problem Conditions for weighted solitons to extend to nearby deformations of Fano manifolds.
method Analyzes the Kuranishi family of Fano manifolds and uses equivariant automorphism groups.
result All members of the Kuranishi family of a Fano manifold with a weighted soliton have weighted solitons if and only if the dimensions of their T-equivariant automorphism groups are equal to that of the original manifold.
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.
We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent sub…
New proof for curvature and diameter estimates on Fano manifolds.
problem Curvature and diameter estimates for Kähler-Ricci flow on Fano manifolds.
method New Harnack estimate for special functions in space-time.
result Established new estimates for scalar curvature and diameter.
The paper defines a new condition for Fano manifolds and shows its implications on their asymptotic behavior.
problem Understanding the asymptotic behavior of Fano manifolds.
method Introducing the asymptotically Mittag-Leffler condition and proving its implications on the -function.
result The -function of a Fano manifold exhibits exponential growth if it is asymptotically Mittag-Leffler.
Study shows volume limit for K-semistable Fano manifolds.
problem Determining the volume of K-semistable Fano manifolds.
method New connection between K-semistability and minimal rational curves.
result Anti-canonical volume is at most for K-semistable Fano manifolds.
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.
Existence of Kähler-Ricci soliton on smooth Fano horospherical manifolds proven.
problem Existence of Kähler-Ricci solitons on smooth Fano horospherical manifolds.
method Study of the Kahler-Ricci flow and convergence in the sense of Cheeger Gromov.
result Renormalized Kahler Ricci flow converges to a Kähler-Ricci soliton.
Study Mabuchi metrics on Fano manifolds proving their existence and properness.
problem Existence and properness of Mabuchi metrics on Fano manifolds.
method Prove existence using properness of modified Ding functional and inverse implication.
result Establish criterion for Mabuchi metrics existence on Fano group compactifications.
Paper studies Kähler-Ricci flow convergence on Fano manifolds.
problem Uniform convergence of Kähler-Ricci flow on Fano manifolds.
method Analyzes flow behavior with varied initial metrics and complex structures.
result Proves uniqueness of Kähler-Ricci solitons in diffeomorphism orbits.
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.
Kähler-Ricci flow shows type II singularity on Fano threefolds.
problem Understanding the behavior of Kähler-Ricci flow on Fano threefolds.
method Analyzing the Kähler-Ricci flow on Fano threefolds from a specific family.
result Kähler-Ricci flow develops type II singularity on Fano threefolds from the specified family.
The Kähler-Ricci flow yields bounds on optimal degenerations and Fano manifolds.
problem Bounding optimal degenerations and Fano manifolds using the Kähler-Ricci flow.
method Using the Kähler-Ricci flow to establish bounds on the Donaldson-Futaki invariant.
result Lower bound for Donaldson-Futaki invariant of optimal degenerations.
We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.