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

TL;DR

This paper constructs special Lagrangian submanifolds in collapsing K3-fibered Calabi-Yau 3-folds, confirming conjectures about their tropicalization and providing insights into mirror symmetry and related conjectures.

Contribution

It provides the first explicit construction of special Lagrangians in collapsing Calabi-Yau 3-folds with K3 fibrations, confirming key conjectures in the field.

Findings

Special Lagrangians shrink to 1-dimensional graphs in the base during collapse.

Results confirm the tropicalization conjecture for holomorphic curves.

Supports the Thomas-Yau and Donaldson-Scaduto conjectures in the Calabi-Yau context.

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.