Calabi-Yau threefolds with positive second Chern class.

*(English)*Zbl 0938.14022From the introduction: D. R. Morrison [in: Journées géométrie algébrique, 1992, Astérisque 218, 243-271 (1993; Zbl 0824.14007)] raised the

Cone conjecture. In the nef cone \(\overline {\text{Amp}}(X)\) of a Calabi-Yau threefold \(X\), there exists a finite rational polyhedral cone \(\Pi\) such that the union of the translates \(\gamma(\Pi)\) by automorphisms \(\gamma\in\operatorname{Aut}(X)\) covers the convex hull of the rational points lying in the nef cone \(\overline{\text{Amp}}(X)\).

However, at the present time, this conjecture has been checked only for one non-trivial example and very little are known in general. On the other hand, in the course of his classification program of Calabi-Yau threefolds according to the behaviour of the second Chern class, P. M. H. Wilson [in: Classification of algebraic varieties, Conf. L’Aquila 1992, Contemp. Math. 162, 403-410 (1994; Zbl 0823.14027)] observed that the automorphism group of a Calabi-Yau threefold \(X\), which is in general a discrete group, is actually finite if the second Chern class is positive, that is, \(c_2(X) \cdot C>0\) for all non-zero nef divisors \(D\), and then asked:

Question 1. Is the nef cone of a Calabi-Yau threefold with positive second Chern class a finite rational polyhedral cone?

Question 2. Are there only finitely many different algebraic fiber space structures on a given Calabi-Yau threefold with positive second Chern class?

Main theorem 1. Every Calabi-Yau threefold with positive second Chern class admits only finitely many proper algebraic fiber space structures, that is, such a Calabi-Yau threefold has only finitely many different \(K3\) fibrations and elliptic fibrations.

It is known that there is a Calabi-Yau threefold (with non-positive second Chern class) which has infinitely many different \(K3\) fibrations and elliptic fibrations [K. Oguiso, Int. J. Math. 4, No. 3, 439-465 (1993; Zbl 0793.14030)]. We also give some partial results concerning the finiteness of birational contractions (propositions 4.1 and 4.6). These together with Wilson’s observation quoted before provide another evidence for the cone conjecture. Our proof of the main theorem 1 (§3) is based on the following topics:

(1) V. Alexeev’s theory on log surfaces [Int. J. Math. 5, No. 6, 779-810 (1994; Zbl 0838.14028)];

(2) Estimates of global log canonical indices of the base spaces of certain elliptic fiber spaces, after Kodaira, Ueno, Kawamata and Nakayama.

Main theorem 2. Every smooth Calabi-Yau threefold in a smooth Fano fourfold has positive second Chern class. In particular, any smooth Calabi-Yau threefolds in a Fano fourfold do not admit abelian fibrations.

Our proof is an application of the following remarkable theorem due to Kollár and several structure theorems of fibered Calabi-Yau threefolds:

Theorem: Let \(V\) be a Fano manifold with \(n=\dim V\geq 4\). Let \(X\in|-K_V|\) be a smooth member. Then the inclusion map \(i_*:\overline{NE}(X)\to\overline{NE}(V)\) is bijective. In particular, the ample cone of a smooth Calabi-Yau threefold in a Fano fourfold is a finite rational polyhedral cone.

Combining these with Wilson’s observation [loc. cit.], we also obtain:

Corollary. The automorphism group of a smooth Calabi-Yau threefold in a Fano fourfold is finite.

Cone conjecture. In the nef cone \(\overline {\text{Amp}}(X)\) of a Calabi-Yau threefold \(X\), there exists a finite rational polyhedral cone \(\Pi\) such that the union of the translates \(\gamma(\Pi)\) by automorphisms \(\gamma\in\operatorname{Aut}(X)\) covers the convex hull of the rational points lying in the nef cone \(\overline{\text{Amp}}(X)\).

However, at the present time, this conjecture has been checked only for one non-trivial example and very little are known in general. On the other hand, in the course of his classification program of Calabi-Yau threefolds according to the behaviour of the second Chern class, P. M. H. Wilson [in: Classification of algebraic varieties, Conf. L’Aquila 1992, Contemp. Math. 162, 403-410 (1994; Zbl 0823.14027)] observed that the automorphism group of a Calabi-Yau threefold \(X\), which is in general a discrete group, is actually finite if the second Chern class is positive, that is, \(c_2(X) \cdot C>0\) for all non-zero nef divisors \(D\), and then asked:

Question 1. Is the nef cone of a Calabi-Yau threefold with positive second Chern class a finite rational polyhedral cone?

Question 2. Are there only finitely many different algebraic fiber space structures on a given Calabi-Yau threefold with positive second Chern class?

Main theorem 1. Every Calabi-Yau threefold with positive second Chern class admits only finitely many proper algebraic fiber space structures, that is, such a Calabi-Yau threefold has only finitely many different \(K3\) fibrations and elliptic fibrations.

It is known that there is a Calabi-Yau threefold (with non-positive second Chern class) which has infinitely many different \(K3\) fibrations and elliptic fibrations [K. Oguiso, Int. J. Math. 4, No. 3, 439-465 (1993; Zbl 0793.14030)]. We also give some partial results concerning the finiteness of birational contractions (propositions 4.1 and 4.6). These together with Wilson’s observation quoted before provide another evidence for the cone conjecture. Our proof of the main theorem 1 (§3) is based on the following topics:

(1) V. Alexeev’s theory on log surfaces [Int. J. Math. 5, No. 6, 779-810 (1994; Zbl 0838.14028)];

(2) Estimates of global log canonical indices of the base spaces of certain elliptic fiber spaces, after Kodaira, Ueno, Kawamata and Nakayama.

Main theorem 2. Every smooth Calabi-Yau threefold in a smooth Fano fourfold has positive second Chern class. In particular, any smooth Calabi-Yau threefolds in a Fano fourfold do not admit abelian fibrations.

Our proof is an application of the following remarkable theorem due to Kollár and several structure theorems of fibered Calabi-Yau threefolds:

Theorem: Let \(V\) be a Fano manifold with \(n=\dim V\geq 4\). Let \(X\in|-K_V|\) be a smooth member. Then the inclusion map \(i_*:\overline{NE}(X)\to\overline{NE}(V)\) is bijective. In particular, the ample cone of a smooth Calabi-Yau threefold in a Fano fourfold is a finite rational polyhedral cone.

Combining these with Wilson’s observation [loc. cit.], we also obtain:

Corollary. The automorphism group of a smooth Calabi-Yau threefold in a Fano fourfold is finite.

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14J50 | Automorphisms of surfaces and higher-dimensional varieties |

57R20 | Characteristic classes and numbers in differential topology |

14D06 | Fibrations, degenerations in algebraic geometry |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14J30 | \(3\)-folds |

14J45 | Fano varieties |

14J35 | \(4\)-folds |