Irregular manifolds whose canonical system is composed of a pencil.

*(English)*Zbl 1075.14038Let \(X\) be a complex projective \(n\)-dimensional manifold of general type whose canonical system is composed with a pencil. Up to replace \(X\) with some blow-up, the canonical map factors through a surjective morphism \(f:X \to C\) onto a curve \(C\) with connected fibers, called the canonical fibration of \(X\). In this setting with \(n=2\), G. Xiao proved that \(q(X) \leq 2\), equality implying that \(C \cong \mathbb P^1\) [Compos. Math. 56, 251–257 (1985; Zbl 0594.14029)].

The paper under review contains some generalizations to higher dimensions. In particular the authors prove the following results.

1) Assume that \(X\) has Albanese dimension \(a(X)=n\). Then, if \(q(X) > n\) then \(p_g(X)=g(C) \geq 2\), \(q(X)=g(C)+n-1\), and the general fiber \(F\) of \(f\) has \(p_g(F)=1\); if \(q(X)=n\), then \(C \cong \mathbb P^1\).

2) Assume that \(q(X) > \text{min}\{a(X)+1,n\}\) and that the image of \(X\) in its Albanese variety has Kodaira dimension \(1\). Then \(p_g(F)=1\) again for the general fiber \(F\) of \(f\). Moreover, \(f\) factors through the Albanese map of \(X\), and \(p_g(X)+1 \geq g(C) \geq 2\), \(q(X)=g(C)+a(X)-1\).

3) The same assertion on the factorization of \(f\) holds for \(n=3\), provided that \(q(X)\geq 5\). The authors show that this bound is the best possible, producing appropriate examples; however they prove that the same property holds also for \(q(X)=4\) and \(3\) under some extra conditions.

The paper under review contains some generalizations to higher dimensions. In particular the authors prove the following results.

1) Assume that \(X\) has Albanese dimension \(a(X)=n\). Then, if \(q(X) > n\) then \(p_g(X)=g(C) \geq 2\), \(q(X)=g(C)+n-1\), and the general fiber \(F\) of \(f\) has \(p_g(F)=1\); if \(q(X)=n\), then \(C \cong \mathbb P^1\).

2) Assume that \(q(X) > \text{min}\{a(X)+1,n\}\) and that the image of \(X\) in its Albanese variety has Kodaira dimension \(1\). Then \(p_g(F)=1\) again for the general fiber \(F\) of \(f\). Moreover, \(f\) factors through the Albanese map of \(X\), and \(p_g(X)+1 \geq g(C) \geq 2\), \(q(X)=g(C)+a(X)-1\).

3) The same assertion on the factorization of \(f\) holds for \(n=3\), provided that \(q(X)\geq 5\). The authors show that this bound is the best possible, producing appropriate examples; however they prove that the same property holds also for \(q(X)=4\) and \(3\) under some extra conditions.

Reviewer: Antonio Lanteri (Milano)

##### MSC:

14J40 | \(n\)-folds (\(n>4\)) |

14J30 | \(3\)-folds |

14J29 | Surfaces of general type |

14K20 | Analytic theory of abelian varieties; abelian integrals and differentials |

14K12 | Subvarieties of abelian varieties |

14D06 | Fibrations, degenerations in algebraic geometry |