Infinite time singularities of the Kähler-Ricci flow

Abstract

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for collapsing limits of Ricci-flat Kahler metrics.

22 pages

References in corpus (5)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• Collapsing of abelian fibred Calabi-Yau manifolds
• The Collapsing Rate of the Kähler-Ricci Flow with Regular Infinite Time Singularity
• Degenerations of Calabi-Yau metrics
• Calabi-Yau manifolds and their degenerations

Cited by in corpus (13)

• KAWA lecture notes on the Kähler-Ricci flow
• The Anomaly flow over Riemann surfaces
• Pluripotential kahler-ricci flows
• Geometry of twisted Kähler-Einstein metrics and collapsing
• Smooth asymptotics for collapsing Calabi-Yau metrics
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Leafwise flat forms on Inoue-Bombieri surfaces
• Curvature Estimates for the Continuity Method
• Collapsing immortal Kähler-Ricci flows
• Generic regularity of intermediate complex structure limits
• Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows
• Almost Non-positive Kähler Manifolds
• Ricci-flat metrics on Calabi-Yau manifolds