Asymptotics for resolutions and smoothings of Calabi-Yau conifolds

Abstract: 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 K\"ahler metrics that are approximately Calabi-Yau. Solving formally a complex Monge-Amp`ere equation, we obtain a polyhomogeneous family of K\"ahler 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.