Degenerations, fibrations and higher rank Landau-Ginzburg models

TL;DR

This paper investigates semi-stable degenerations of quasi-Fano varieties and proposes a conjecture that higher rank Landau-Ginzburg models can be glued to produce models mirror to the original varieties, supported by various invariants and periods.

Contribution

It introduces a conjecture relating higher rank Landau-Ginzburg models to degenerations of quasi-Fano varieties and proves it through invariants and period comparisons.

Findings

Proved the conjecture relating Landau-Ginzburg models and degenerations.

Established relations between Euler characteristics and functional invariants.

Constructed higher codimension Calabi-Yau fibrations.

Abstract

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are mirror to the original quasi-Fano varieties. We prove this conjecture by relating their Euler characteristics, generalized functional invariants as well as periods. We also use it to conjecture a relation between the degenerations to the normal cones and the fibrewise compactifications of higher rank Landau-Ginzburg models. Furthermore, we use it to iterate the Doran-Harder-Thompson conjecture and obtain higher codimension Calabi-Yau fibrations.