Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds
arXiv:1012.2940
Abstract
In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat Kähler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interests.
arXiv commentAn appendix is written by Mark Gross
References in corpus (8)
- Limits of Calabi-Yau metrics when the Kahler class degenerates
- Geometric Transitions
- The Kahler-Ricci flow through singularities
- The Kähler-Ricci flow on Hirzebruch surfaces
- Balanced metrics on non-Kahler Calabi-Yau threefolds
- Metric flips with Calabi ansatz
- Collapsing of Calabi-Yau manifolds and special lagrangian submanifolds
- Convergence of Calabi-Yau manifolds
Cited by in corpus (6)
- The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
- Gromov-Hausdorff collapsing of Calabi-Yau manifolds
- Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
- Families of Calabi-Yau manifolds and canonical singularities
- Nakamaye's theorem on complex manifolds
- On the Kahler Ricci flow on projective manifolds of general type