  Let Y be a projective non-singular curve of genus g, X a projective manifold,
both defined over the field of complex numbers, and let f:X ---> Y be a
surjective morphism with general fibre F.
  If the Kodaira dimension of X is non-negative, and if Y is the projective
line we show that f has at least 3 singular fibres.
  In general, for non-isotrivial morphisms f, one expects that the number of
singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong
additivity of the Kodaira dimension, this is verified, if either F is of
general type, or if F has a minimal model with a semi-ample canonical divisor.
The corresponding result has been obtained by Migliorini and Kovacs, for
families of surfaces of general type and for families of canonically polarized
manifolds, and by Oguiso-Viehweg for families of elliptic surfaces.
  As a byproduct we obtain explicit bounds for the degree of the direct image
of powers of the dualizing sheaf, generalizing those obtained by
Bedulev-Viehweg for families of surfaces of general type.
