turkmath.org

Schoen has shown that a fiber product of two relatively minimal rational elliptic surfaces with section is a simply connected Calabi-Yau 3-fold if the fiber product is smooth and the same is true for the desingularization of the fiber product by small resolutions in the case that the singularities are ordinary double points. I will describe the small resolution process and talk about lifting automorphisms on the fiber product to automorphisms of the desingularization. I will discuss the problem of constructing fixed point free finite group actions on such desingularizations. The quotient of the 3-fold by such group actions give rise to non-simply connected Calabi-Yau 3-folds. The problem on the existence of such group actions on smooth fiber products was solved by previous works of Bouchard, Donagi and the speaker.