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

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.