Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces

§ 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:Being more optimistic, one might ask:Conjecture C;i,m' If IC(W);;;::O then Var (f) is defined to be the minimal number k, such that there exists a subfield L of C( W) of transcendental degree k over C and a variety F over L with FXSpec(Ll Spec (C(W)),.,.. Vw (.,... means "birational").