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<p>We show that the Calabi–Yau metrics with isolated conical singularities of Hein-Sun (Publ Math de l’IHÉS, 126(1):73–130, 2017) 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.</p>

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Asymptotics for Resolutions and Smoothings of Calabi-Yau Conifolds

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Abdou Oussama Benabida

摘要

We show that the Calabi–Yau metrics with isolated conical singularities of Hein-Sun (Publ Math de l’IHÉS, 126(1):73–130, 2017) 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.