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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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