Fibers over infinity of Landau-Ginzburg models

TL;DR

This paper explores a conjecture relating the components of fibers over infinity in Landau-Ginzburg models to the anticanonical system dimension of smooth Fano varieties, verified in several cases including toric and Calabi-Yau compactifications.

Contribution

It proposes a new conjecture connecting fiber components over infinity with the anticanonical system dimension and verifies it for various classes of Fano varieties.

Findings

Conjecture holds for toric Landau-Ginzburg models of Fano threefolds

Verified for complete intersections and some toric varieties

Supports a deeper link between Landau-Ginzburg fibers and Fano geometry

Abstract

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.