  Let f: V --> U be a smooth non-isotrivial family of canonically polarized
n-dimensional complex manifolds, where U is the complement of a normal crossing
divisor S in a projective manifold Y. We show that some symmetric product of
the sheaf of one-forms with logarithmic poles along S contains an invertible
subsheaf of positive Kodaira dimension. As a corollary one finds that U can not
be a complete intersection in the projective N space of codimension l < N/2,
nor the complement of l general hyperplanes, for l < N. Moreover, as shown by
S. Kovacs before, U can not be a projective manifold with a nef tangent bundle.
  If the induced morphism to the moduli scheme is generically finite, one can
also exclude U to be the product of more than n curves. Moreover, the existence
of the family forces the automorphism group of U to be finite.
  Most of those results carry over to smooth families f:V --> U, with an
f-semi-ample dualizing sheaf, provided f is of maximal variation.
