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On algebraic fiber space structures on a Calabi-Yau 3-fold. Appendix by Noburu Nakayama. (English) Zbl 0793.14030
In 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.
Reviewer: A.Iliev (Sofia)

MSC:
14J30 \(3\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
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