Toric mirror monodromies and Lagrangian spheres

TL;DR

This paper explores the relationship between toric degenerations and mirror symmetry, demonstrating that line bundles correspond to Lagrangian spheres in the Calabi-Yau case, advancing understanding of homological mirror symmetry.

Contribution

It shows that in the Calabi-Yau case, line bundles are represented by Lagrangian spheres within the framework of toric mirror symmetry.

Findings

Line bundles correspond to Lagrangian spheres in the Calabi-Yau case.

Homological mirror symmetry holds for the central fiber of toric degenerations.

The work connects algebraic and symplectic geometry through mirror symmetry.

Abstract

The central fiber of a Gross-Siebert type toric degeneration is known to satisfy homological mirror symmetry: its category of coherent sheaves is equivalent to the wrapped Fukaya category of a certain exact symplectic manifold. Here we show that, in the Calabi-Yau case, the images of line bundles are represented by Lagrangian spheres.