Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

Details Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

2010-12-14

arxiv.org/abs/1012.2940v2

Mathematics

Differential Geometry

An appendix is written by Mark Gross

Scientific paper

In this paper, we study the behavior of Ricci-flat K\"{a}hler 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 K\"{a}hler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interests.

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