Non-simply connected Calabi–Yau threefolds constructed as quotients of Schoen threefolds

Abstract

The aim of this paper is to complete the classification of all Calabi–Yau threefolds which are constructed as the quotient of a smooth Schoen threefold X = B 1 × P 1 B 2 (fiber product over P 1 of two relatively minimal rational elliptic surfaces B 1 and B 2 with section) under a finite group action acting freely on the Schoen threefold X . The abelian group actions on smooth Schoen threefolds which induce cyclic group actions on the base curve P 1 were studied by Bouchard and Donagi (2008), and all such actions were listed. We consider the actions on the Schoen threefold by finite groups G whose elements are given as a product τ 1 × τ 2 of two automorphisms τ 1 and τ 2 of the rational elliptic surfaces B 1 and B 2 with section. In this paper, we use the classification of automorphism groups of rational elliptic surfaces with section given in Karayayla (2012) and Karayayla (2014) to generalize the results of Bouchard and Donagi to answer the question whether finite and freely acting group actions on Schoen threefolds which induce non-cyclic group actions on the base curve P 1 exist or not. Despite the existence of group actions on rational elliptic surfaces which induce non-cyclic (even non-abelian) group actions on P 1 , it is shown in this paper that none of those actions can be lifted to free actions on a Schoen threefold. The main result is that there is no finite group action on a Schoen threefold X which acts freely on X and which induces a non-cyclic group action on the base curve P 1 . This result shows that the list given in Bouchard and Donagi (2008) is a complete list of non-simply connected Calabi–Yau threefolds constructed as the quotient of a smooth Schoen threefold by a finite group action.