Collapsing of abelian fibred Calabi-Yau manifolds

Abstract

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting we prove an analog of a result of Gross-Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.

30 pages; shortened exposition and corrected typos; final version to appear in Duke Math. J

References in corpus (4)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• Base manifolds for fibrations of projective irreducible symplectic manifolds
• Degenerations of Calabi-Yau metrics
• Geometric Structures of Collapsing Riemannian Manifolds I

Cited by in corpus (19)

• KAWA lecture notes on the Kähler-Ricci flow
• The Chern-Ricci flow on complex surfaces
• Remarks on the collapsing of torus fibered Calabi-Yau manifolds
• Infinite time singularities of the Kähler-Ricci flow
• Collapsing of the Chern-Ricci flow on elliptic surfaces
• Families of Calabi-Yau manifolds and canonical singularities
• Smooth and Rough Positive Currents
• Geometry of twisted Kähler-Einstein metrics and collapsing
• Smooth asymptotics for collapsing Calabi-Yau metrics
• Non-archimedean integrals as limits of complex integrals
• Ricci-flat metrics and dynamics on K3 surfaces
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Curvature Estimates for the Continuity Method
• Leafwise flat forms on Inoue-Bombieri surfaces
• Survey on the metric SYZ conjecture and non-archimedean geometry
• Collapsing immortal Kähler-Ricci flows
• Ricci-flat metrics on Calabi-Yau manifolds
• Almost Non-positive Kähler Manifolds
• On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows