Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

TL;DR

This paper proves that smooth Fano threefolds possess toric Landau-Ginzburg models, providing explicit constructions and describing their fiber structures, thereby advancing the understanding of mirror symmetry in algebraic geometry.

Contribution

It establishes the existence of toric Landau-Ginzburg models for smooth Fano threefolds and constructs explicit models for del Pezzo surfaces and their divisors.

Findings

Landau-Ginzburg models admit compactifications to K3 surface families

Explicit models for del Pezzo surfaces are provided

Fibers over infinity are characterized

Abstract

We prove that smooth Fano threefolds have toric Landau--Ginzburg models. More precise, we prove that their Landau--Ginzburg models, presented as Laurent polynomials, admit compactifications to families of K3 surfaces, and we describe their fibers over infinity. We also give an explicit construction of Landau--Ginzburg models for del Pezzo surfaces and any divisors on them.