Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections

TL;DR

This paper investigates Landau-Ginzburg models for Fano threefolds with Picard rank one, comparing their singular fiber data to known exceptional collections, and explores implications for Homological Mirror Symmetry.

Contribution

It provides a detailed analysis of singular fibers in Landau-Ginzburg models and relates this to exceptional collections and mirror symmetry predictions for Fano threefolds.

Findings

Comparison of fiber data with known exceptional collection lengths

Verification of some Homological Mirror Symmetry predictions

Formulation of expectations for exceptional collections in Fano threefolds

Abstract

We study fibers with isolated singularities of Landau-Ginzburg models for Fano threefolds of Picard rank one. We compare the data we get with maximal known lengths of exceptional collections in derived categories of coherent sheaves on the Fano threefolds, verify some predictions of Homological Mirror Symmetry, and present some expectations about exceptional collections for Fano threefolds.