Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models

TL;DR

This paper demonstrates that all rank one smooth Fano threefolds and certain higher-dimensional Fano varieties admit weak Landau-Ginzburg models derived from toric degenerations, with fibers compactifying to K3 surfaces.

Contribution

It introduces a method to construct weak Landau-Ginzburg models for Fano threefolds and higher-dimensional Fano varieties via toric degenerations, linking them to K3 surfaces.

Findings

Weak Landau-Ginzburg models exist for all rank one smooth Fano threefolds.

Fibers of these models can be compactified to K3 surfaces with Picard rank 19.

Fano complete intersections in weighted projective spaces also admit such models.

Abstract

We show that every rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau-Ginzburg model coming from a toric degeneration.