On a class of non-simply connected Calabi-Yau $3$-folds with positive Euler characteristic

In this work we obtain a class of non-simply connected Calabi–Yau $3$ folds with positive Euler characteristic as the quotient of projective small resolutions of singular Schoen $3$ folds under the free action of finite groups. A Schoen $3$ fold is a fiber product $X = B_1 \times {}_{\mathbb{P}^1} \: B_2$ of two relatively minimal rational elliptic surfaces with section $\beta_i : B_i \to \mathbb{P}^1 , i = {1, 2}$. Schoen has shown that if $X$ is smooth, then $X$ is a simply connected Calabi–Yau $3$ fold, and if the only singularities of $X$ are on $\mathbb{I}_r \times \mathbb{I}_s$ type fibers with $r \gt 1$ and $s \gt 1$, then there exists a projective small resolution $\hat{X}$ of $X$, and $\hat{X}$ is a simply connected Calabi–Yau $3$ fold [7]. If $G$ is a finite group which acts freely on a smooth Schoen $3$ fold $X$, then the quotient $X/G$ is a non-simply connected Calabi–Yau $3$ fold with fundamental group $G$, and all such group actions have been classified by Bouchard and Donagi [2]. Bouchard and Donagi have proposed the open problem of classifying all finite groups $G$ which act freely on projective small resolutions $\hat{X}$ of singular Schoen $3$ folds $X$. In this case the quotient $\hat{X}/G$ is again a Calabi–Yau $3$ fold with fundamental group $G$. In this paper we first classify the finite groups $G$ which act freely on singular Schoen $3$ folds $X$ where the only singularities of $X$ are on $\mathbb{I}_r \times \mathbb{I}_s$ type fibers with $r \gt 1$ and $s \gt 1$ and the elements of $G$ act on $X$ as an automorphism $\tau_1 \times \tau_2$ where each $\tau_i$ is an automorphism of the elliptic surface $B_i$. A projective small resolution $\hat{X}$ of $X$ is obtained by blowing up some components of the $\mathbb{I}_r \times \mathbb{I}_s$ fibers on $X$. We determine which of the free actions on the singular $3$‑fold $X$ lift to free actions on the Calabi–Yau $3$ fold $\hat{X}$. For the non-simply connected Calabi–Yau $3$ folds $\hat{X}/G$ obtained with this construction, the distinct fundamental groups are $\mathbb{Z}_3 \times \mathbb{Z}_3$, $\mathbb{Z}_4 \times \mathbb{Z}_2$, $\mathbb{Z}_2 \times \mathbb{Z}_2$, and $\mathbb{Z}_n$ for $n = 6, 5, 4, 3, 2$. These are the same groups obtained by Bouchard and Donagi by working on free actions on smooth Schoen $3$ folds. While the Euler characteristic of each $X/G$ obtained by Bouchard and Donagi is $0$, the Euler characteristics of all non-simply connected Calabi–Yau $3$ folds $\hat{X}/G$ we obtain in this paper are positive and they range in: $64$, $54$, $48$, $40$ and $2k$ for $2 \leq k \leq 18$. The given Euler characteristic values do not all occur for each of the listed fundamental groups. The classification of finite groups which act freely on singular Schoen $3$ folds $X$ whose singularities are on $\mathbb{I}_r \times \mathbb{I}_s$ type fibers with $r \gt 1$ and $s \gt 1$, the classification of such group actions which lift to free actions on projective small resolutions $\hat{X}$ of $X$, and the fundamental groups and Euler characteristic values of the non-simply connected Calabi–Yau $3$ folds $\hat{X }/G$ are displayed in several tables. The study of the group actions on $X$ which induce a non-trivial action on the base curve $\mathbb{P}^1$ and which induce a trivial action on $\mathbb{P}^1$ is carried out separately.

Keywords

Calabi–Yau $3$-folds, Schoen $3$-folds, fiber product of relatively minimal rational elliptic surfaces with section, nonsimply connected Calabi–Yau $3$-folds, group actionss, automorphisms of rational elliptic surfaces

2010 Mathematics Subject Classification

14J27, 14J30, 14J32, 14J50, 14L30

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This work has been supported by Middle East Technical University Scientific Research Projects Coordination Unit under grant number GAP-101-2018-2789.

Received 15 September 2020

Accepted 17 December 2021

Published 1 February 2022