The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits

Abstract

We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kahler metrics.

42 pages, final version to appear in Amer. J. Math

References in corpus (9)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• The Kahler-Ricci flow through singularities
• Remarks on the collapsing of torus fibered Calabi-Yau manifolds
• Riemannian geometry of Kahler-Einstein currents
• Ricci flow and birational surgery
• Collapsing of Products Along the Kähler-Ricci Flow
• Degenerations of Calabi-Yau metrics
• Soliton-type metrics and Kähler-Ricci flow on symplectic quotients
• Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

Cited by in corpus (23)

• KAWA lecture notes on the Kähler-Ricci flow
• Inoue surfaces and the Chern-Ricci flow
• The Anomaly flow over Riemann surfaces
• Relative volume comparison of Ricci Flow and its applications
• The Chern-Ricci flow on Oeljeklaus-Toma manifolds
• Bounding diameter of singular Kähler metric
• Smooth and Rough Positive Currents
• On collapsing Calabi-Yau fibrations
• The continuity method on minimal elliptic Kähler surfaces
• Geometry of twisted Kähler-Einstein metrics and collapsing
• A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry
• Smooth asymptotics for collapsing Calabi-Yau metrics
• The continuity method on Fano fibrations
• Geometric estimates for complex Monge-Ampere equations
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension
• Curvature Estimates for the Continuity Method
• Degeneration of Ricci-flat Calabi-Yau manifolds and its applications
• Collapsing immortal Kähler-Ricci flows
• Ricci-flat metrics on Calabi-Yau manifolds
• The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds
• On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows
• Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows