On Arithmetic Mirror Symmetry for smooth Fano fourfolds

TL;DR

The paper constructs explicit classes of Laurent polynomials as Landau--Ginzburg models for smooth Fano fourfolds, exploring their implications for Arithmetic Mirror Symmetry and providing concrete examples of mirror correspondences.

Contribution

It introduces explicit tempered Laurent polynomials for Fano fourfolds and investigates their role in Arithmetic Mirror Symmetry, including new mirror symmetry examples.

Findings

Contains Landau--Ginzburg models for various Fano fourfolds.

Supports the Arithmetic Mirror Symmetry conjecture in specific cases.

Constructs examples of mirror symmetry correspondence between algebraic classes.

Abstract

We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in $n \leqslant 4$ variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, or are quiver flag zero loci. We discuss implications to Arithmetic Mirror Symmetry conjecture, a Hodge-theoretic approach to the study of Ap\'{e}ry constants of Fano varieties proposed by Golyshev--Kerr--Sasaki. Using the partial case of Arithmetic Mirror Symmetry conjecture proved by Kerr, we construct two examples of a Mirror Symmetry correspondence between specific algebraic classes.