IRREGULAR MANIFOLDS WHOSE CANONICAL SYSTEM IS COMPOSED OF A PENCIL

Abstract

Let X be a complex projective n-dimensional manifold of general type whose canonical system is composite with a pencil. If the Albanese map is generically finite, but not surjective, or if the irregularity is strictly larger than n and the image of X in Alb(X) is of Kodaira dimension one, then the geometric genus p_g(F) of a general fibre F of the canonical map is one and the latter factors through the Albanese map. The last part of this result holds true for any threefold with q(X) ≥ 5.