Threefolds with vanishing Hodge cohomology

We investigate a question raised by J.-P. Serre in 1953: Let Y be a complex manifold with Hi( Y, WjY ) = 0 for all j ≥ 0 and i > 0 (where WjY is the sheaf of holomorphic j-forms on Y ), then what is Y? Is Y Stein? In 1991, T. Peternell [P] proved that if Y is also holomorphically convex, then Y is Stein. And when Y is an analytic surface, he gave some sufficient condition for Steinness. At the same time, he asked the same question for smooth varieties: If Y is a smooth variety over C , is it affine? Suppose Y is an algebraic manifold, i.e., an irreducible nonsingular algebraic variety defined over C . If dimY = 1, then Y is affine. If dimY = 2, N. Mohan Kumar classified it completely. Roughly speaking, if Y is not affine, then it is a surface obtained from a special ruled surface or special rational and ruled surface by removing a special divisor. What will happen when dimension is 3? If there are nonconstant regular functions on Y, we will understand it by looking at its smooth completion X and the fibre space from X to a smooth projective curve. In this case, Y is a fibre space over a smooth affine curve C (i.e., we have a surjective morphism from Y to C such that general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. In fact, we prove that the smooth fibres are deformation invariant, i.e., all smooth fibres must be of the same type. Generally, if Y is not affine, then the Kodaira dimension of X is -infinity and the D-dimension is 1, where D is an effective divisor of X supported at X - Y with normal crossings. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of Y. Our basic tools are local cohomology theory and classification theory of higher dimensional varieties developed by the Japanese school of algebraic geometry.