A Finiteness Theorem for Elliptic Calabi-Yau Threefolds

The analagous class for surfaces are the K3 surfaces. All K3 surfaces are homeomorphic: there is one underlying topological type. On the other hand, there are a large number of topological types of minimal Calabi-Yau threefolds, but it is an open question of whether there are a finite number of such types. A stronger question would be to ask whether there are a finite number of families of algebraic minimal Calabi-Yau threefolds. This is definitely not true for K3 surfaces: there are a countably infinite number of algebraic families. Up to birational equivalence, we answer this question for those Calabi-Yaus which possess an elliptic fibration.