Berglund-H\"ubsch-Krawitz Mirrors via Shioda Maps

TL;DR

This paper presents a new elementary proof of the birationality of multiple Berglund-H"ubsch-Krawitz mirrors using Shioda maps, avoiding toric geometry and unnecessary assumptions.

Contribution

It introduces a novel, elementary approach to proving the birationality of BHK mirrors via Shioda maps, providing explicit birational models.

Findings

BHK mirrors are birational via explicit quotients of Fermat varieties.

The proof avoids toric geometry and simplifies previous methods.

Provides a general birational picture of the BHK correspondence.

Abstract

We give an elementary approach to proving the birationality of multiple Berglund-H\"ubsch-Krawitz (BHK) mirrors by using Shioda maps. We do this by creating a birational picture of the BHK correspondence in general. Although a similar result has been obtained in recent months by Shoemaker, our proof is new in that it sidesteps using toric geometry and drops an unnecessary hypothesis. We give an explicit quotient of a Fermat variety to which the mirrors are birational.