Riemannian geometry of Kahler-Einstein currents

Abstract

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 projective variety, with well-defined tangent cones. We also prove a special degeneration for Kahler-Einstein manifolds of general type as an approach to establish the compactification of the moduli space of Kahler-Einstein manifolds of general type. A number of applications are given for degeneration of Calabi-Yau manifolds and the Kahler-Ricci flow on smooth minimal models of general type.

References in corpus (11)

• K-stability and Kähler-Einstein metrics
• Limits of Calabi-Yau metrics when the Kahler class degenerates
• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• The Kahler-Ricci flow through singularities
• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors
• Ricci flow and birational surgery
• Degeneration of Kahler-Ricci solitons on Fano manifolds
• Bounding scalar curvature for global solutions of the Kahler-Ricci flow
• A remark on Kahler metrics with conical singularities along a simple normal crossing divisor
• On the structure of almost Einstein manifolds

Cited by in corpus (24)

• The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• On the Yau-Tian-Donaldson conjecture for singular Fano varieties
• Relative volume comparison of Ricci Flow and its applications
• Collapsing hyperkähler manifolds
• Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow
• Degeneration of Kahler-Einstein manifolds of negative scalar curvature
• Riemannian geometry of Kahler-Einstein currents II: an analytic proof of Kawamata's base point free theorem
• Bounding diameter of singular Kähler metric
• The local entropy along Ricci flow---Part A: the no-local-collapsing theorems
• Convergence of Kähler-Ricci flow on lower dimensional algebraic manifolds of general type
• Riemannian geometry of Kahler-Einstein currents III: compactness of Kahler-Einstein manifolds of negative scalar curvature
• On the Kahler Ricci flow on projective manifolds of general type
• A remark on constant scalar curvature Kähler metrics on minimal models
• Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow
• A continuity method to construct canonical metrics
• Twisted Kähler-Einstein Metrics and Collapsing
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Geometric estimates for complex Monge-Ampere equations
• Kahler-Einstein metric near an isolated log canonical singularity
• Degeneration of Ricci-flat Calabi-Yau manifolds and its applications
• Existence of Kahler-Ricci solitons on smoothable Q-Fano varities
• Bounding diameter of conical Kahler metric
• The continuity equation with cusp singularities