On the cone of divisors of Calabi-Yau fiber spaces.

*(English)*Zbl 0931.14022A 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.

(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.

Reviewer: Chen Zhijie (Shanghai)