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Special Lagrangian submanifolds in K3-fibered Calabi-Yau 3-folds

Shih-Kai Chiu, Yu-Shen Lin

Abstract

We construct special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces. As these 3-folds collapse, the special Lagrangians shrink to 1-dimensional graphs in the base, mirroring the conjectured tropicalization of holomorphic curves in collapsing SYZ torus-fibered Calabi-Yau manifolds. This confirms predictions of Donaldson and Donaldson-Scaduto in the Calabi-Yau setting. Additionally, we discuss our results in the contexts of the Thomas-Yau conjecture, the Donaldson-Scaduto conjecture, and mirror symmetry.

Special Lagrangian submanifolds in K3-fibered Calabi-Yau 3-folds

Abstract

We construct special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces. As these 3-folds collapse, the special Lagrangians shrink to 1-dimensional graphs in the base, mirroring the conjectured tropicalization of holomorphic curves in collapsing SYZ torus-fibered Calabi-Yau manifolds. This confirms predictions of Donaldson and Donaldson-Scaduto in the Calabi-Yau setting. Additionally, we discuss our results in the contexts of the Thomas-Yau conjecture, the Donaldson-Scaduto conjecture, and mirror symmetry.

Paper Structure

This paper contains 30 sections, 50 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. Special Lagrangian submanifolds
  3. Fix loci of anti-holomorphic, anit-symplectic involutions
  4. Hyperkähler rotations
  5. Deformations of special Lagrangian submanifolds
  6. Calabi-Yau metrics on K3-fibered Calabi-Yau 3-folds
  7. Calabi-Yau metrics on $\mathbb{C}^3$
  8. The approximate solutions
  9. Weighted spaces on $X$
  10. Deforming to genuine Calabi-Yau metrics
  11. Analysis on the special Lagrangian thimble
  12. Geometry of the special Lagrangian thimble
  13. Weighted spaces
  14. Analysis on cylinders
  15. Approximating the Green function
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\gamma$ be an admissible path with phase $\theta$ connecting two points $y_0, y_1$ in the discriminant locus $S \subset Y$. Then for sufficiently small $t > 0$, there exists a special Lagrangian $3$-sphere $\tilde{L}_{\gamma, t}$ with phase $\theta$ in the Calabi-Yau manifold $(X, \tilde{\omega

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 91 more