A holomorphic line bundle L on a complex space X is said to be nef if deg(L\(| C)\geq 0\) for every compact complex curve C in X; if X is compact, L is said to be big if, for each irreducible component Y of X, \(L| Y\) has Kodaira dimension dim(Y). In this paper, the author proves the following vanishing theorem: Let \(f: X\to Y\) be a proper morphism between irreducible complex spaces, with X smooth. Let L in Pic(X) be such that \(i()\quad L\) is nef on each fibre of f, and \((ii)\quad there\) is a second category subset V of Y such that L is big on each \(f^{- 1}(y)\), \(y\in U\). Then R \(qf_*(K_ X\otimes L)=0\) for \(q\geq 1\). He deduces that if (ii) holds with L replaced by \(pL-K_ X\) for some integer \(p>0\), then the natural map f \(*f_*({\mathcal O}_ X(mL))\to {\mathcal O}_ X(mL)\) is surjective on each compact subset E of Y, for \(m\geq m_ 0(E).\)
These results are relative complex-analytic analogues of results of Ramanujam, Kawamata, Viehweg, and Shokurov in the algebraic case. See also the author and A. Silva, Lect. Notes Math. 1165, 1-13 (1985; Zbl 0583.32065).