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.
Survey article on Kahler-Einstein metrics and algebraic geometry.
problem Understanding Kahler-Einstein metrics in algebraic geometry.
method Not specified in the abstract, likely involves mathematical analysis and algebraic geometry techniques.
result Discussion of recent developments and challenges in the field.
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.
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.
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.
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.
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. 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.
Explains Kähler-Einstein metrics and algebraic structures.
problem None explicitly stated in the abstract.
method Expository lecture.
result None explicitly stated in the abstract.
Extends Faltings heights to arithmetic varieties and connects to Kahler-Einstein geometry.
problem Connecting arithmetic heights to Kahler-Einstein geometry.
method Extending Faltings heights, proposing arithmetic Yau-Tian-Donaldson conjecture.
result Direct relations between arithmetic heights and Kahler-Einstein geometry.
Study shows Kähler-Einstein metric singularities linked to curvature.
problem Understanding singularities of Kähler-Einstein metrics.
method Relates singularities to holomorphic sectional curvature of conical geometry.
result Provides second-order estimates with explicit constants.
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.
We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original project…
The study of real Einstein submanifolds in Kähler geometry.
problem Understanding real Einstein submanifolds in Kähler geometry.
method Provided a necessary and sufficient condition for anti-holomorphic automorphisms to determine real Einstein submanifolds.
result A condition for determining real Einstein submanifolds in compact Kähler-Einstein manifolds.
Futaki invariant vanishes on Hopf manifolds.
problem Obtaining Kähler-Einstein metrics on compact manifolds.
method Generalized Futaki invariant to Hopf manifolds and proved its vanishing.
result Futaki invariant vanishes on Hopf manifolds.
Smooth Kahler-Einstein metrics have been studied for the past 80 years. More recently, singular Kahler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a surve…
The paper explores the geometry of real connections on Hermitian manifolds and derives Kähler-Einstein metrics.
problem Investigating the geometry of real connections on Hermitian manifolds.
method Analyzing the relationship between real connections and Hermitian connections, focusing on the real Chern connection.
result Derives Kähler-Einstein metrics using real Chern-Einstein metrics.
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.
Different distances on symmetrical domains in complex space.
problem Comparing distances on specific complex domains.
method Examined Carathéodory pseudo-distance and Kähler-Einstein metric distances.
result Found the distances differ on certain complex domains.
Study on cusp singularities in Kähler-Einstein metrics.
problem Analyzing cusp singularities in Kähler-Einstein metrics.
method Continuity method applied to varieties with cusp singularities.
result Investigation of differential and algebro-geometric properties of the limit.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. In this short note, we show that given a special Kähler-Einstein degeneration with bounded geometry, for any noncentral fiber, there exists a Kähler-Ricci flow which converges to the Kähler-Einstein metric of the central fiber. As an example, Tian's deformations \cite{Tian97} of the Mukai 3-fold admit Kähler-Ricci flow…
New metrics found on Hirzebruch surfaces solve complex geometry questions.
problem Finding new conformally Kähler, Einstein-Maxwell metrics on Hirzebruch surfaces.
method Used special Killing potentials found by Futaki and Ono.
result Proved existence of new conformally Kähler, Einstein-Maxwell metrics.
New invariants prove existence of Kahler-Einstein metrics on big classes.
problem Existence of Kahler-Einstein metrics on varieties with klt singularities.
method Introducing new invariants and proving a generalization of the Tian-Odaka-Sano Theorem.
result Proves existence of twisted Kahler-Einstein metrics on big classes.
Tian's conjectures solved in Kahler geometry, linking metrics and inequalities.
problem Existence of Kahler-Einstein metrics and other canonical metrics.
method Analytic characterization using Sobolev inequalities.
result Strong Moser-Trudinger inequalities for Kahler geometry.
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.
Flow proves Kähler-Einstein on minimal elliptic surfaces.
problem Proving Kähler-Einstein on minimal elliptic surfaces.
method Conical Kähler-Ricci flow with semi-ampleness assumption.
result Flow converges to Kähler-Einstein on canonical model.
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.
Study geodesics in totally real submanifolds of Kähler-Einstein manifolds.
problem Geodesics in totally real submanifolds of Kähler-Einstein manifolds.
method Define a canonical connection and geodesics on the space of totally real submanifolds, and study their properties.
result Existence and uniqueness of geodesics in totally real submanifolds.
Proves existence of Kähler-Einstein metric on K-stable Fano manifolds.
problem Existence of Kähler-Einstein metrics on Fano manifolds.
method Compactness of Kähler-Ricci flows and algebro-geometric description of their asymptotics.
result Existence of Kähler-Einstein metric on K-stable Fano manifolds.
The paper constructs almost para-Kähler-Einstein metrics on cotangent bundles.
problem Developing metrics on cotangent bundles associated with geometric structures.
method Using a construction involving geometric structures on a manifold M to associate an almost para-Kähler-Einstein metric on T∗M. result Explicit formulae for these metrics are derived in specific geometric cases.
We calculate an upper bound for the second nonzero eigenvalue of the scalar Laplacian, λ2, for toric Kähler-Einstein metrics in terms of the polytope data. We give some detailed examples in complex dimensions 1, 2 and 3. We also discuss extensions of this method to other geometries.
Study of twisted Kähler-Einstein metrics on Calabi-Yau spaces with singularities.
problem Understanding the collapsed Gromov-Hausdorff limits of Calabi-Yau spaces.
method Analyzing the geometry of twisted Kähler-Einstein metrics on holomorphic fiber spaces.
result Proving the existence of conical-type singularities in the base of fiber spaces.
The scalar curvature is redefined in generalized Kahler geometry as a moment map.
problem Defining scalar curvature in generalized Kahler geometry.
method Introducing a moment map in generalized Kahler geometry to define a generalized scalar curvature.
result Infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite-dimensional.
The study explores Einstein metrics in (2,3,5) distributions and their geometric connections.
problem Existence of Einstein metrics in conformal structures induced by (2,3,5) distributions.
method Characterization of conformal structures that admit almost Einstein scales and use of holonomy reduction.
result Established novel links between (2,3,5) distributions and various geometries, including Sasaki-Einstein and Kähler-Einstein.
New stability concept tested via volume function for Fano manifolds.
problem Stability of Fano manifolds and existence of Kähler-Einstein metrics.
method Introducing and testing divisorial stability via volume functions.
result Existence of Kähler-Einstein metrics is equivalent to divisorial semistability for toric Fano manifolds.
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 …
Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
problem Understanding the asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.
method Analyzing the curvature and metric properties of Kähler-Einstein metrics on complex hyperbolic cusps.
result Sharp doubly exponential rate of convergence of metrics to a limiting form.
Flow analysis leads to metric completion in Kähler geometry.
problem Analyzing Kähler-Ricci flows on compact manifolds.
method Normalized Kähler-Ricci flow convergence to Gromov-Hausdorff limits.
result Metric completion of twisted Kähler-Einstein metric.
Survey explains Kahler-Einstein metrics construction from interpolation problems.
problem Kahler-Einstein metrics on compact complex manifolds.
method Statistical mechanical construction from interpolation problems.
result Probabilistic construction of Kahler solutions to Einstein's equations.
Study on geometric properties of plurisubharmonic functions in strongly pseudoconvex domains.
problem Metric properties and regularity of Mabuchi geodesics in the space of strongly plurisubharmonic functions.
method Introduction of Mabuchi space, study of metric properties using Mabuchi geodesics, establishment of regularity properties.
result Existence of local Kähler-Einstein metrics as an application.
Study on Kähler-Einstein metrics on log Calabi-Yau families, proving local triviality and discrediting a metric conjecture.
problem Understanding Kähler-Einstein metrics on log Calabi-Yau families.
method Investigation of relative Ricci-flat Kähler metrics, focusing on Kodaira dimension zero.
result Disproof of a folklore conjecture about the semipositivity of relative flat metrics.
New probabilistic method constructs Kähler-Einstein metrics and suggests zero-free properties of zeta functions.
problem Existence and explicit formulas for Kähler-Einstein metrics on Fano varieties.
method Probabilistic construction involving canonical random point processes.
result Zero-free properties of Archimedean zeta functions and their relation to Langlands program.
Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
problem Characterize compact quasi-Einstein metrics with constant scalar curvature.
method Connection to Sasakian geometry and circle bundles over Einstein metrics.
result Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.
Compact Kahler-Einstein manifolds converge to semi-log canonical models.
problem Compactness of Kahler-Einstein manifolds of negative scalar curvature.
method Gromov-Hausdorff convergence and Weil-Petersson metric extension.
result Convergence to a finite union of complete Kahler-Einstein metric spaces.
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and L2-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Rie…