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Gromov-Hausdorff continuity of non-Kähler Calabi-Yau conifold transitions

Benjamin Friedman, Sébastien Picard, Caleb Suan

TL;DR

The paper addresses how to realize a GH-continuous interpolation between Calabi–Yau spaces connected by a conifold transition, even as Hodge numbers jump and non-Kähler geometries arise. It develops a robust framework built on balanced and Hermitian–Yang–Mills metrics, combining local Calabi–Yau models (Candelas–de la Ossa) with Fu–Li–Yau gluing to control global geometry across small resolutions and smoothings. Central technical achievements include precise diameter/volume estimates near singular loci, a curve-reduction Main Lemma for GH control, and careful convergence analysis of both the metric and Yang–Mills data. The results establish GH-convergence of the SR and smoothing geometries to the singular base, providing a mathematically rigorous non-Kähler extension of conifold-transition continuity with potential implications for moduli and string theory. Overall, the work offers a comprehensive, quantitative pathway for passing through topological transitions while preserving metric-geometric continuity in a non-Kähler Calabi–Yau context.

Abstract

We study the geometry of Calabi-Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian-Yang-Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov-Hausdorff topology.

Gromov-Hausdorff continuity of non-Kähler Calabi-Yau conifold transitions

TL;DR

The paper addresses how to realize a GH-continuous interpolation between Calabi–Yau spaces connected by a conifold transition, even as Hodge numbers jump and non-Kähler geometries arise. It develops a robust framework built on balanced and Hermitian–Yang–Mills metrics, combining local Calabi–Yau models (Candelas–de la Ossa) with Fu–Li–Yau gluing to control global geometry across small resolutions and smoothings. Central technical achievements include precise diameter/volume estimates near singular loci, a curve-reduction Main Lemma for GH control, and careful convergence analysis of both the metric and Yang–Mills data. The results establish GH-convergence of the SR and smoothing geometries to the singular base, providing a mathematically rigorous non-Kähler extension of conifold-transition continuity with potential implications for moduli and string theory. Overall, the work offers a comprehensive, quantitative pathway for passing through topological transitions while preserving metric-geometric continuity in a non-Kähler Calabi–Yau context.

Abstract

We study the geometry of Calabi-Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian-Yang-Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov-Hausdorff topology.
Paper Structure (33 sections, 30 theorems, 215 equations, 1 figure)

This paper contains 33 sections, 30 theorems, 215 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Gromov--Hausdorff Topology
  4. Conifold Transitions
  5. Metrics on Small Resolutions
  6. Candelas--de la Ossa Metrics on the Local Model
  7. Balanced Metrics on the Small Resolution
  8. Hermitian--Yang--Mills Metrics on the Small Resolution
  9. Metrics on Smoothings
  10. Candelas--de la Ossa Metrics on the Local Model
  11. Balanced Metrics on the Smoothings
  12. Hermitian--Yang--Mills Metrics on the Smoothing
  13. Notation and conventions
  14. Gromov--Hausdorff Continuity in the Regular Case
  15. Gromov--Hausdorff vs Uniform Convergence
  16. ...and 18 more sections

Key Result

Theorem 1.3

Let $\widehat{X}$ be a compact Kähler Calabi--Yau threefold with finite fundamental group. Let $\widehat{X} \rightarrow X_0 \rightsquigarrow X_t$ be a conifold transition. There exists a family of smooth metrics $( \widehat{X} , \widehat{g} _a, \widehat{H} _a)$ for $0<a<1$ and $(X_t,g_t,H_t)$ for $0 such that as the parameters $a$ and $t$ are varied, the geometries $(X, \widehat{g} _a, \widehat{H}

Figures (1)

  • Figure 1: Local model of a conifold transition.

Theorems & Definitions (54)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: R. Friedman Fri86Fri91
  • Definition 2.7
  • ...and 44 more