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On vanishing of higher direct images of the structure sheaf

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arxiv 2410.15282 v3 pith:ZOJJNNL5 submitted 2024-10-20 math.AG math.AC

Shihoko Ishii , Ken-ichi Yoshida This is my paper · ORCID
classification math.AGmath.AC
keywords schemedirectdimensionimagenormalregularsheafstructure
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abstract

We show the vanishing of the first direct image of the structure sheaf of a normal scheme $X$ which is mapped properly and birationally over a regular scheme of any dimension. On the other hand, for any dimension greater than two, we show examples of a proper birational morphism from a normal and Cohen-Macaulay scheme to a regular scheme such that the second direct image does not vanish and has an isolated support.

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  1. On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type

    math.AG 2025-06 conditional novelty 7.0 of 10

    The paper proves degree-one Grauert-Riemenschneider vanishing for Cohen-Macaulay klt-type schemes and, in dimension three, full GR vanishing and rational singularities.

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