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Threefolds with vanishing Hodge cohomology

Author: Jing Zhang
Journal: Trans. Amer. Math. Soc. 357 (2005), 1977-1994
MSC (2000): Primary 14J30, 14B15, 14C20
Published electronically: October 7, 2004
MathSciNet review: 2115086
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Abstract: 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 $Y$.

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Additional Information

Jing Zhang
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Address at time of publication: Department of Mathematics, University of Missouri–Columbia, Columbia, Missouri 65211

Keywords: 3-folds, Hodge cohomology, local cohomology, fibration, higher direct images
Received by editor(s): May 9, 2003
Received by editor(s) in revised form: November 21, 2003
Published electronically: October 7, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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