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Irregular manifolds whose canonical system is composed of a pencil. (English) Zbl 1075.14038
Let \(X\) 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.

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
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