Mirror Quintics, discrete symmetries and Shioda Maps

TL;DR

This paper provides a simple proof that certain families of mirror quintic threefolds share the same Picard-Fuchs equation, and relates quotients of these families to mirror quintics using Shioda maps, with generalizations to higher dimensions.

Contribution

It offers an easy argument for the shared Picard-Fuchs equation and relates quotient families to mirror quintics via Shioda maps, extending to higher-dimensional Calabi-Yau varieties.

Findings

All six families have the same Picard-Fuchs equation.

Quotients of these families relate to the Mirror Quintics.

Generalizations to degree n Calabi-Yau varieties are established.

Abstract

In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard Fuchs equation associated to the holomorphic 3-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one parameter families to the family of Mirror Quintics. Our constructions generalize to degree n Calabi Yau varieties in (n-1)-dimensional projective space.