Complex structure degenerations and collapsing of Calabi-Yau metrics
Complex structure degenerations and collapsing of Calabi-Yau metrics
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Calabi-Yau 度量的复杂结构退化和崩溃
DOI：
10.1090/s0894-0347-1990-1040196-6
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Ruobing Zhang
中科院分区：
文献类型：
--
作者：
Song Sun;Ruobing Zhang
在本文中，我们对退化的极化Calabi-Yau流形族上的Calabi-Yau度量的坍缩行为的理解取得了进展。在射影空间中一类光滑Calabi-Yau超曲面一般退化为两个光滑Fano超曲面的横向并的情况下，我们得到了在所有维度上完备的结果，建立了度量坍缩与复杂结构退化之间明确而精确的关系。这个结果即使在复维二维中也是新的。这是通过胶合和奇异摄动技术实现的，以及一个关键的几何成分，涉及到具有环面对称的某些（不一定是光滑的）Kahler度量的构造。我们还讨论了将此结果扩展到更一般设置的可能性。
In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way, we obtain a complete result in all dimensions establishing explicit and precise relationships between the metric collapsing and complex structure degenerations. This result is new even in complex dimension two. This is achieved via gluing and singular perturbation techniques, and a key geometric ingredient involving the construction of certain (not necessarily smooth) Kahler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings.