Calabi-Yau threefolds of quasi-product type

  • Keiji Oguiso

    Department of Mathematical Sciences University of Tokyo Komaba, Tokyo Japan
Calabi-Yau threefolds of quasi-product type cover
Download PDF

This article is published open access.


According to the numerical Iitaka dimension ν(X,D)\nu(X,D) and c2(X)Dc_{2}(X) \cdot D, fibered Calabi-Yau threefolds ΦD:XW(dimW>0)\Phi_{|D|} : X \rightarrow W ({dim } W > 0) are coarsely classified into six different classes. Among these six classes, there are two peculiar classes called of type II0{II}_{0} and of type III0{III}_{0} which are characterized respectively by ν(X,D)=2\nu(X,D) = 2 and c2(X)D=0c_{2}(X) \cdot D = 0 and by ν(X,D)=3\nu(X,D) = 3 and c2(X)D=0c_{2}(X) \cdot D = 0. Fibered Calabi-Yau threefolds of type III0{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 II0{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.

Cite this article

Keiji Oguiso, Calabi-Yau threefolds of quasi-product type. Doc. Math. 1 (1996), pp. 417–447

DOI 10.4171/DM/18