Asymptotics for resolutions and smoothings of Calabi-Yau conifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914308601937920 |
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| author | Benabida, Abdou Oussama |
| author_facet | Benabida, Abdou Oussama |
| contents | We show that the Calabi-Yau metrics with isolated conical singularities of Hein-Sun admit polyhomogeneous expansions near their singularities. Moreover, we show that, under certain generic assumptions, natural families of smooth Calabi-Yau metrics on crepant resolutions and on polarized smoothings of conical Calabi-Yau manifolds degenerating to the initial conical Calabi-Yau metric admit polyhomogeneous expansions where the singularities are forming. The construction proceeds by performing weighted Melrose-type blow-ups and then gluing conical and scaled asymptotically conical Calabi-Yau metrics on the fibers, close to the blow-up's front face without compromising polyhomogeneity. This yields a polyhomogeneous family of Kähler metrics that are approximately Calabi-Yau. Solving formally a complex Monge-Ampère equation, we obtain a polyhomogeneous family of Kähler metrics with Ricci potential converging rapidly to zero as the family is degenerating. We can then conclude that the corresponding family of degenerating Calabi-Yau metrics is polyhomogeneous by using a fixed point argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics for resolutions and smoothings of Calabi-Yau conifolds Benabida, Abdou Oussama Differential Geometry Mathematical Physics Algebraic Geometry Analysis of PDEs 32Q25, 53C55 We show that the Calabi-Yau metrics with isolated conical singularities of Hein-Sun admit polyhomogeneous expansions near their singularities. Moreover, we show that, under certain generic assumptions, natural families of smooth Calabi-Yau metrics on crepant resolutions and on polarized smoothings of conical Calabi-Yau manifolds degenerating to the initial conical Calabi-Yau metric admit polyhomogeneous expansions where the singularities are forming. The construction proceeds by performing weighted Melrose-type blow-ups and then gluing conical and scaled asymptotically conical Calabi-Yau metrics on the fibers, close to the blow-up's front face without compromising polyhomogeneity. This yields a polyhomogeneous family of Kähler metrics that are approximately Calabi-Yau. Solving formally a complex Monge-Ampère equation, we obtain a polyhomogeneous family of Kähler metrics with Ricci potential converging rapidly to zero as the family is degenerating. We can then conclude that the corresponding family of degenerating Calabi-Yau metrics is polyhomogeneous by using a fixed point argument. |
| title | Asymptotics for resolutions and smoothings of Calabi-Yau conifolds |
| topic | Differential Geometry Mathematical Physics Algebraic Geometry Analysis of PDEs 32Q25, 53C55 |
| url | https://arxiv.org/abs/2503.16702 |