Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds

By Xiaochun Rong, Yuguang Zhang. 2010

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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.

View Paper: https://www.semanticscholar.org/paper/b95c45b981517f23ac551f3c30d493a147b6df85

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Citations in the corpus

Gromov-Hausdorff collapsing of Calabi-Yau manifolds (2013, result): “In the case of non-collapsing limits, analogous results about metric completions have been obtained in [25, 26, 27], and homeomorphism results have been obtained in [6] (see also [33]).”
On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, methodology): “Based on uniform L∞-estimate obtained in Step 3, we apply the gauge fixing technique used in [45, 47, 42] and prove that (X(0,t), d(0,t)) is the metric completion of the strong conical Kähler-metric structure (M \ E, (1 − t)mH,ω(0,t)).”
On the Kahler Ricci flow on projective manifolds of general type (2015, unknown): “1 in [16]), there exists a surjective map (RX , g∞) → (X∞, d∞),”
Collapsing of abelian fibred Calabi-Yau manifolds (2011, background): “The general question of understanding the geometric behaviour of these metrics was raised by Yau [46, 47], Wilson [44] and others, and much work has been devoted to it, see for example [18, 29, 30, 34, 37, 38] and references therein.”
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014, background): “By comparison, in the case when the volume of the Ricci-flat metrics is non-collapsing, we have a very thorough understanding of their limiting behavior, thanks to the results in [8, 31, 32, 33, 37, 53], and all the analogous questions have affirmative answers.”