Automorphisms and quotients of Calabi-Yau threefolds of type $A$

TL;DR

This paper classifies automorphisms and quotients of specific Calabi-Yau threefolds derived from abelian threefolds, providing new constructions, classifications, and topological invariants of resulting Calabi-Yau varieties.

Contribution

It offers a complete classification of automorphism groups and quotients of two families of Calabi-Yau threefolds of type A, including new constructions and topological invariants.

Findings

Classified automorphism groups of the Calabi-Yau threefolds in the families.

Constructed and classified quotients of these threefolds under subgroup actions.

Computed Hodge numbers and fundamental groups of desingularized quotients.

Abstract

The aim of the paper is to investigate the only two families $\mathcal{F}^A_{G}$ of Calabi-Yau $3$-folds $A/G$ with $A$ an abelian $3$-fold and $G\le \text{Aut}(A)$ a finite group acting freely: one in constructed by Catanese and Demleitner and the other is presented here. We provide a complete classification of the automorphism group of $X\in \mathcal{F}^A_{G}$. Additionally, we construct and classify the quotients $X/\Upsilon$ for any $\Upsilon\le \text{Aut}(X)$. Specifically, for those groups $\Upsilon$ that preserve the volume form of $X$ then $X/\Upsilon$ admits a desingularization $Y$ which is a Calabi-Yau $3$-fold: we compute the Hodge numbers and the fundamental group of these $Y$, thereby determining all topological in-equivalent Calabi-Yau $3$-folds obtained in this way.