On singular log Calabi-Yau compactifications of Landau-Ginzburg models

TL;DR

This paper explores the construction of singular log Calabi-Yau compactifications for Landau-Ginzburg models of Fano varieties, revealing singularities and confirming a conjecture relating fiber components to the anticanonical system.

Contribution

It provides new insights into the singularities of compactifications for certain Landau-Ginzburg models and verifies a conjecture connecting fiber components with the Fano variety's anticanonical system.

Findings

For degree greater than 2 coverings, the compactification is singular.

No smooth projective log Calabi-Yau compactification exists for these cases.

The conjecture relating fiber components to the anticanonical system holds in the studied cases.

Abstract

We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it for del Pezzo surfaces and coverings of projective spaces of index one. For the coverings of degree greater then 2 the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove in the cases under consideration the conjecture saying that the number of components of the fiber over infinity %and of finite fibers is equal to the dimension of an anticanonical system of the Fano variety.