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Complex structure degenerations and collapsing of Calabi-Yau metrics

Song Sun, Ruobing Zhang

Abstract

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way, we obtain a complete result in all dimensions establishing explicit and precise relationships between the metric collapsing and complex structure degenerations. This result is new even in complex dimension two. This is achieved via gluing and singular perturbation techniques, and a key geometric ingredient involving the construction of certain (not necessarily smooth) Kähler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings.

Complex structure degenerations and collapsing of Calabi-Yau metrics

Abstract

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way, we obtain a complete result in all dimensions establishing explicit and precise relationships between the metric collapsing and complex structure degenerations. This result is new even in complex dimension two. This is achieved via gluing and singular perturbation techniques, and a key geometric ingredient involving the construction of certain (not necessarily smooth) Kähler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings.

Paper Structure

This paper contains 46 sections, 75 theorems, 804 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Outline of the proof of the main theorem
  4. Organization of the paper
  5. Acknowledgements
  6. Calabi-Yau metrics with torus symmetry
  7. Motivation and dimension reduction of the Calabi-Yau equation
  8. Calabi model spaces
  9. Two dimensional standard model spaces
  10. Linearized equation and singularities
  11. Higher rank torus symmetry
  12. Green's currents
  13. Normal coordinates for an embedded submanifold
  14. Green's currents for Riemannian submanifolds
  15. Green's currents on a cylinder
  16. ...and 31 more sections

Key Result

Theorem 1.2

The following statements hold:

Figures (5)

  • Figure 1.1: The algebraic family $\mathcal{X}$
  • Figure 1.2: The collapsing Calabi-Yau metric $\omega_{CY,t}$ on $X_t$
  • Figure 4.1: Subdivision of $\mathcal{M}_T$ into various regions
  • Figure 6.1: The modified family $\widehat{\mathcal{X}}$
  • Figure 6.2: Division of a neighborhood of $\widehat{X}_0$

Theorems & Definitions (180)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 3.1: Normal coordinates
  • Lemma 3.2: Generalized Gauss Lemma
  • Definition 3.3: Normal regularity order
  • Example 3.4
  • Remark 3.5
  • ...and 170 more