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Threefolds with vanishing Hodge cohomology. (English) Zbl 1067.14033
Summary: We consider algebraic manifolds \(Y\) of dimension 3 over \(\mathbb{C}\) with \(H^i(Y, \Omega^j_Y)=0\) for all \(j\geq 0\) and \(i>0\). Let \(X\) be a smooth completion of \(Y\) with \(D=X-Y\), an effective divisor on \(X\) with normal crossings. If the \(D\)-dimension of \(X\) is not zero, then \(Y\) is a fibre space over a smooth affine curve \(C\) (i.e., we have a surjective morphism from \(Y\) to \(C\) 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 \(X\) is \(-\infty\) and the \(D\)-dimension of \(X\) 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 \(Y\).

MSC:
14J30 \(3\)-folds
14B15 Local cohomology and algebraic geometry
14F40 de Rham cohomology and algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C20 Divisors, linear systems, invertible sheaves
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