Degenerations and Lagrangian fibrations of Calabi-Yau manifolds

TL;DR

This paper explores degenerations and Lagrangian fibrations of Calabi-Yau manifolds, focusing on Tyurin degenerations and the Doran-Harder-Thompson conjecture, linking mirror symmetry concepts and proposing new Landau-Ginzburg models.

Contribution

It advances understanding of mirror symmetry by proving the Doran-Harder-Thompson conjecture and introduces a novel Landau-Ginzburg model construction.

Findings

Proof of the Doran-Harder-Thompson conjecture.

New insights into Tyurin degenerations and mirror symmetry.

Proposal of a new Landau-Ginzburg model construction.

Abstract

We discuss various topics on degenerations and special Lagrangian torus fibrations of Calabi-Yau manifolds in the context of mirror symmetry. A particular emphasis is on Tyurin degenerations and the Doran-Harder-Thompson conjecture, which builds a bridge between mirror symmetry for Calabi-Yau manifolds and that for quasi-Fano manifolds. The proof of the conjecture is of interest in its own right and leads us to a few other related topics such as SYZ mirror symmetry, theta functions and geometric quantization. Inspired by the conjecture, we also propose a new construction of Landau-Ginzburg models by splitting Calabi-Yau fibrations.