Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections
Victor Przyjalkowski
TL;DR
The paper investigates Landau–Ginzburg models for Fano threefolds of Picard rank one through the lens of Homological Mirror Symmetry, connecting the singular fibers of Calabi–Yau compactifications to exceptional collections in derived categories. It defines and analyzes invariants $\varepsilon(X)$, $\omega(X)$, and $\omega_{\mathcal{O}}(X)$ to compare the Fukaya–Seidel side with algebraic geometry, proving explicit counts of ordinary double points for the LG models associated to each family and identifying cases where these counts determine the maximal length of exceptional collections. The results show that, for many families, the maximal length of exceptional collections matches the number of ODP fibers, while highlighting special cases (e.g., $X_3$ and certain non-generic deformations) where this correspondence fails or requires refinement. The work also discusses the role of mutations, standard versus Givental-type LG models, and the limitations of relying solely on singularities to predict categorical structure, illustrated by concrete counterexamples. Overall, the paper advances precise links between LG-model geometry and derived-category invariants for Picard rank one Fano threefolds and clarifies when HMS predictions yield full exceptional collections.
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.