Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

TL;DR

This paper studies the suspension operation on Lefschetz fibrations, showing it preserves the Fukaya category and alters the fiber category, and applies this to prove part of Homological Mirror Symmetry for certain toric varieties.

Contribution

It introduces the suspension operation on Lefschetz fibrations and demonstrates its invariance properties, applying this to establish results in Homological Mirror Symmetry.

Findings

Fukaya category remains invariant under suspension

Suspension modifies the fiber's category in a controlled way

Partial proof of Homological Mirror Symmetry for canonical bundles over toric del Pezzo surfaces

Abstract

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.