Calabi-Yau threefolds of quasi-product type

Abstract

According to the numerical Iitaka dimension $\nu (X,D)$ and $c_2(X)\cdot D$, fibered Calabi-Yau threefolds ${\Phi }_{|D|}:X\to W(dimW>0)$ are coarsely classified into six different classes. Among these six classes, there are two peculiar classes called of type ${II}_0$ and of type ${III}_0$ which are characterized respectively by $\nu (X,D)=2$ and $c_2(X)\cdot D=0$ and by $\nu (X,D)=3$ and $c_2(X)\cdot D=0$. Fibered Calabi-Yau threefolds of type ${III}_0$ are intensively studied by Shepherd-Barron, Wilson and the author and now there are a satisfactory structure theorem and the complete classification. The purpose of this paper is to guarantee a complete structure theorem of fibered Calabi-Yau threefolds of type ${II}_0$ to finish the classification of these two peculiar classes. In the course of proof, the log minimal model program for threefolds established by Shokurov and Kawamata will play an important role. We shall also introduce a notion of quasi-product threefolds and show their structure theorem. This is a generalization of the notion of hyperelliptic surfaces to threefolds and will have other applicability, too.