Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

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.

An appendix is written by Mark Gross

References in corpus (3)

• The Kahler-Ricci flow through singularities
• Metric flips with Calabi ansatz
• Convergence of Calabi-Yau manifolds

Cited by in corpus (2)

• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• On the Kahler Ricci flow on projective manifolds of general type