On the finiteness of fiber-space structures on a Calabi-Yau 3-fold. (English. Russian original) Zbl 1087.14509

Oguiso, Keiji

J. Math. Sci., New York 106, No. 5, 3320-3335 (2001); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 70, 134-158 (2001).

From the introduction: This is a survey of recent progress in fiber-space structures on a Calabi-Yau 3-fold. The first three sections are quite introductory. The last three sections are devoted to a discussion of some finiteness properties of fiber-space structures.
Throughout this article, we emphasize geometric aspects and illustrate how one can apply the general theory of 3-folds, namely, log-minimal model theory, its basic idea, and the theory of surfaces to study Calabi-Yau 3-folds. Also, several properties of Calabi-Yau 3-folds related to this problem are discussed sometimes via examples and sometimes by showing an idea of proof. Several unsolved questions are also included.

MSC:

14J32	Calabi-Yau manifolds (algebro-geometric aspects)
14E30	Minimal model program (Mori theory, extremal rays)
14J30	\(3\)-folds