Gromov-Hausdorff collapsing of Calabi-Yau manifolds

Abstract

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau manifolds are Lagrangian fibrations of holomorphic symplectic manifolds, the metrics on the regular parts of the limits are special Kähler metrics. By combining these two results, we extend arXiv:math/0008018 to any fibered projective K3 surface without any assumption on the type of singular fibers.

References in corpus (4)

• Base manifolds for fibrations of projective irreducible symplectic manifolds
• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Degenerations of Calabi-Yau metrics
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Cited by in corpus (28)

• The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
• Families of Calabi-Yau manifolds and canonical singularities
• Higher-order estimates for collapsing Calabi-Yau metrics
• Collapsing hyperkähler manifolds
• Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow
• Triviality of fibered Calabi-Yau manifolds without singular fibers
• Nakamaye's theorem on complex manifolds
• Special Lagrangian submanifolds of log Calabi-Yau manifolds
• Ergodic complex structures on hyperkahler manifolds
• Canonical bundle formula and degenerating families of volume forms
• Collapsing Ricci-flat metrics on elliptic K3 surfaces
• PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics
• Geometry of twisted Kähler-Einstein metrics and collapsing
• A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry
• Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing
• Non-archimedean integrals as limits of complex integrals
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Twisted Kähler-Einstein Metrics and Collapsing
• Gromov-Hausdorff limits of flat Riemannian surfaces
• Special Kähler geometry and holomorphic Lagrangian fibrations
• Degeneration of Ricci-flat Calabi-Yau manifolds and its applications
• The Strominger-Yau-Zaslow conjecture and its impact
• Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces
• Completion of the moduli space for polarized Calabi-Yau manifolds
• Special Kähler structures, cubic differentials and hyperbolic metrics
• Balanced embedding of degenerating Abelian varieties
• On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows
• Ricci-flat metrics on Calabi-Yau manifolds