Base spaces of non-isotrivial families of smooth minimal models

Abstract

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.