COMMUNICATIONS IN NUMBER THEORY AND PHYSICS, vol.16, no.1, pp.159-213, 2022 (Journal Indexed in SCI)
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 x(P1) B-2 of two relatively minimal rational elliptic surfaces with section beta(i) : B-i -> 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 I-r x I-s type fibers with r > 1 and s > 1, then there exists a projective small resolution (X) over cap of X, and (X) over cap 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 (X) over cap of singular Schoen 3-folds X. In this case the quotient (X) over cap /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 I-r x I-s type fibers with r > 1 and s > 1 and the elements of G act on X as an automorphism tau(1) x tau(2) where each tau(i) is an automorphism of the elliptic surface B-i. A projective small resolution (X) over cap of X is obtained by blowing up some components of the I-r x 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 (X) over cap. For the non-simply connected Calabi-Yau 3-folds (X) over cap /G obtained with this construction, the distinct fundamental groups are Z(3) x Z(3), Z(4) x Z(2), Z(2) x Z(2), and 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) over cap /G obtained by Bouchard and Donagi is 0, the Euler characteristics of all non-simply connected Calabi-Yau 3-folds X/G we obtain in this paper are positive and they range in: 64, 54, 48, 40 and 2k for 2 <= k <= 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 I-r x I-s type fibers with r > 1 and s > 1, the classification of such group actions which lift to free actions on projective small resolutions (X) over cap of X, and the fundamental groups and Euler characteristic values of the non-simply connected Calabi-Yau 3-folds (C) over cap /G are displayed in several tables. The study of the group actions on X which induce a non-trivial action on the base curve P-1 and which induce a trivial action on P-1 is carried out separately.