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On the cone of divisors of Calabi-Yau fiber spaces. (English) Zbl 0931.14022
A projective surjective morphism of normal varieties with geometrically connected fibers $$f:X\rightarrow S$$ is called a Calabi-Yau fiber space if $$X$$ has only $$\mathbb{Q}$$-factorial terminal singularities and the canonical divisor $$K_X$$ is relatively trivial over $$S$$. This concept is a natural generalization of that of Calabi-Yau manifolds. In this paper the following generalization of the Morrison conjecture is considered:
(1) The number of the Aut$$(X/S)$$-equivalence classes of faces of the cone $$\mathcal A^e(X/S)$$ corresponding to birational contractions or fiber space structures is finite. Moreover, there exists a finite rational polyhedral cone which is a fundamental domain for the action of $$\operatorname{Aut}(X/S)$$ on $$\mathcal A^e(X/S)$$.
(2) The number of the Bir$$(X/S)$$-equivalence classes of chambers $$\mathcal A^e(X//S,\alpha)$$ in the cone $$\mathcal M^e(X/S)$$ for the marked minimal models $$f':X'\rightarrow S$$ of $$f$$ is finite. Moreover, there exists a finite rational polyhedral cone which is a fundamental domain for the action of Bir$$(X/S)$$ on $$\mathcal M^e(X/S)$$.
It is already known that the above conjectures are true if $$\dim X=\dim S=3$$. In the paper it is proved that the first parts of the conjectures (1) and (2) in the case where $$0<\dim S<\dim X=3$$ are also true.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14E30 Minimal model program (Mori theory, extremal rays) 14C20 Divisors, linear systems, invertible sheaves
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