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Asymptotics for resolutions and smoothings of Calabi-Yau conifolds

Abdou Oussama Benabida

TL;DR

The paper develops a comprehensive framework for polyhomogeneous expansions of Calabi–Yau metrics on conifolds and their degenerations, proving that Hein–Sun conical CY metrics admit polyhomogeneous expansions and that, under generic Assumptions R and S, natural families of CY metrics on crepant resolutions and polarized smoothings are likewise polyhomogeneous as the singularities form. The approach merges weighted Melrose-type blow-ups, gluing of conical and asymptotically conical CY metrics, formal expansion of the complex Monge–Ampère equation, and a fixed-point argument to obtain genuine degenerating families. The results provide precise boundary asymptotics (in the Melrose sense) for Ricci-flat metrics through conifold transitions, with two principal desingularization paradigms—crepant resolutions and polarized smoothings—treated in parallel. This yields a robust, polyhomogeneous description of degenerations that has potential implications for spectral geometry and moduli theory in Calabi–Yau settings, including uniform resolvent constructions and refined convergence notions beyond Gromov–Hausdorff limits.

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 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.

Asymptotics for resolutions and smoothings of Calabi-Yau conifolds

TL;DR

The paper develops a comprehensive framework for polyhomogeneous expansions of Calabi–Yau metrics on conifolds and their degenerations, proving that Hein–Sun conical CY metrics admit polyhomogeneous expansions and that, under generic Assumptions R and S, natural families of CY metrics on crepant resolutions and polarized smoothings are likewise polyhomogeneous as the singularities form. The approach merges weighted Melrose-type blow-ups, gluing of conical and asymptotically conical CY metrics, formal expansion of the complex Monge–Ampère equation, and a fixed-point argument to obtain genuine degenerating families. The results provide precise boundary asymptotics (in the Melrose sense) for Ricci-flat metrics through conifold transitions, with two principal desingularization paradigms—crepant resolutions and polarized smoothings—treated in parallel. This yields a robust, polyhomogeneous description of degenerations that has potential implications for spectral geometry and moduli theory in Calabi–Yau settings, including uniform resolvent constructions and refined convergence notions beyond Gromov–Hausdorff limits.

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 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.

Paper Structure

This paper contains 25 sections, 25 theorems, 128 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Examples and discussion
  5. Organization of the paper
  6. Preliminaries on b-geometry
  7. Manifolds with corners
  8. Function spaces on manifolds with corners
  9. Polyhomogeneous functions
  10. b-fibrations and the push-forward theorem
  11. Blow-up in the sense of Melrose
  12. Conifolds
  13. Analysis on conifolds
  14. Calabi-Yau conifolds
  15. Polyhomgeneity of conical Calabi-Yau manifolds
  16. ...and 10 more sections

Key Result

Theorem A

If $(X_0, \omega_{CY})$ as above is such that $\omega_{CY} \in [\omega] \in H^{1,1}(X_0^{reg}, \mathbb{R})$ for a smoothly Kähler form $\omega$ and $X_0$ has trivial canonical bundle, then the conical Calabi-Yau metric $\omega_{CY}$ admits a polyhomogeneous expansion near every singularity in terms

Figures (3)

  • Figure 1: The blown-up parametric space $\mathcal{M}_b$ in the case of a crepant resolution.
  • Figure 2: Examples of manifolds with corners with codimensions of faces indicated.
  • Figure 3: These are not manifolds with corners. The cone and the pyramid are not modeled on $\mathbb{R}^{n,k}$ near the vertex/apex. The teardrop is a t-manifold but the boundary is not embedded.

Theorems & Definitions (76)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • ...and 66 more