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§ 1 Weak positivity § 2 CoveringsContents § 3 Applications of duality theory § 4 The proof of theorem III for k = 1 § 5 The main lemmata and the proof of theorem III § 6 Products of fibre spaces § 7 The proofs of theorem II and corollary IV § 8 The proof of theorem I § 9 Classification theory Let V and W be non-singular projective varieties over the field of complex numbers C, n= dim (V) and m=dim (W). Let/: V---+W be a fibre space (this simply means that I is surjective with connected general fibre Vw=VXw Spec (C(W)). We denote the canonical sheaves of V and Wby wyand WW, and we write wy, w =wii9l*wy/.S. Iitaka conjectured the following inequality for the Kodaira dimension to be true:

Given a polynomial h of degree n let M h be the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h. By [23] there exist a quasi-projective scheme M h together with a natural transformationsuch that M h is a coarse moduli scheme for M h . For a complex quasi-projective manifold U we will say that a morphism ϕ : U → M h factors through the moduli stack, or that ϕ is induced by a family, if ϕ lies in the image of Ψ(U),Let Y be a projective non-singular compactification of U such that S = Y \U is a normal crossing divisor, and assume that the morphism ϕ : U → M h , induced by a family, is generically finite. For moduli of curves of genus g ≥ 2, i.e. for h(t) = (2t − 1)(g − 1), it is easy to show, that the existence of ϕ forces Ω [25] geometric properties of manifolds U mapping non-trivially to the moduli stack. Again, U can not be C * , nor an abelian variety, and more generally it must be Brody hyperbolic, if ϕ is quasi-finite.In general the sheaf Ω 1 X/Y (log S) fails to be big (see example 6.3). Nevertheless, building up on the methods introduced in [24] and [25] we will show that for m sufficiently large the sheaf S m (Ω 1 Y (log S)) has enough global sections (see section 1 for the precise statement), to exclude the existence of a generically finite morphism ϕ : U → M h , or even of a non-trivial morphism, for certain manifolds U.

The numerical characterization of bad linear subspaces claimed in the proposition of Section 3 is wrong; namely the implication d) ~ b) does not hold true.The following counterexample is due to A. Varchenko. Consider the direct product of two affine arrangements, each of them consisting of n >2 lines on a plane intersecting at one point; then take the projective closure of this arrangement. In this arrangement the singular point satisfies a) but does not satisfy c).The mistake occurs in the first line of p. 561, where we claim that the arrangement H'/= H'ilH'io again satisfies the assumption b), overlooking the possibility that H~'= H] for i+j. Therefore one can not argue inductively, the Euler characteristic being a topological invariant not taking in account the multiplicities. The easy implications a)~--~b) and c)~ b) remain true. The proof of the theorem (and of its corollary) does not use the numerical characterizations c), d) of (Bad) and therefore is not touched.We are grateful to A. Varchenko for drawing our attention to the above example.