On algebraic fiber space structures on a Calabi-Yau 3-fold. Appendix by Noburu Nakayama.

*(English)*Zbl 0793.14030In this paper the author proves a structure theorem for minimal Calabi- Yau threefolds (i.e. \(\mathbb{Q}\)-factorial, normal, projective 3-folds with only terminal singularities, trivial canonical class, \(c_ 2 \neq 0\), and \(\pi_ 1^{\text{alg}}=\{1\})\). More precisely, an appropriate multiple \(m.L\) of any nef and effective divisor \(L\) on a minimal Calabi- Yau threefold \(X\) gives a fibration \(\varphi:X \to W\), and the main theorem in the paper explicitly describes the structure of the general fiber of \(\varphi\), and the type of \(W\), depending on the invariants \(\kappa (X,L)=\nu (X,L)=\dim W\) and \(L.c_ 2 (X)\). In addition, the author constructs the first known example of a Calabi-Yau threefold “of type II\(_ 0\)” (i.e. \(\nu (X,L)=2\), \(L.c_ 2 (X)=0)\). As an application, a simpler, and a little generalized proof of the theorem of Peternell for the existence of a rational curve on a Calabi-Yau threefold, which contains a nonzero and nonample effective divisor, is given.

The appendix, written by N. Nakayama, is devoted to the existence of families of rational curves on certain Calabi-Yau threefolds admitting a fibering structure. In connection with this, a theorem, which describes the structure of such threefolds having no families of rational curves, is proved.

The appendix, written by N. Nakayama, is devoted to the existence of families of rational curves on certain Calabi-Yau threefolds admitting a fibering structure. In connection with this, a theorem, which describes the structure of such threefolds having no families of rational curves, is proved.

Reviewer: A.Iliev (Sofia)