Weak Landau-Ginzburg models for smooth Fano threefolds

TL;DR

This paper constructs explicit Laurent polynomial Landau-Ginzburg models for all smooth Fano threefolds with Picard rank 1, verifying their Calabi-Yau compactifications and analyzing their geometric properties.

Contribution

It provides explicit Laurent polynomial models for all 17 smooth Fano threefolds with Picard rank 1 and studies their compactifications and associated invariants.

Findings

Models can be compactified to open Calabi-Yau varieties

Number of irreducible components relates to intermediate Jacobians

Models are consistent across different compactifications

Abstract

The paper is joined with arXiv:0911.5428 and improved. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactifications. We state most of known methods of finding Landau-Ginzburg models in terms of Laurent polynomials. We discuss Laurent polynomial representation of Landau-Ginzburg models of Fano varieties and state some problems related to it.