Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and . Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. 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 .