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Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

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arxiv 1012.2940 v2 pith:QBGT3DMN submitted 2010-12-14 math.DG math.AGmath.MG

Xiaochun Rong , Yuguang Zhang This is my paper · ORCID
classification math.DGmath.AGmath.MG
keywords calabi-yaumanifoldsricci-flatextremalflopstransitionshlermetrics
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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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  1. Calabi-Yau threefolds across quadratic singularities

    math.DG 2025-01 unverdicted

    This paper is a survey of the geometry of conifold transitions between Calabi-Yau threefolds, focusing on non-Kähler outputs and the structures they carry.

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