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LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES

Part of: Surfaces and higher-dimensional varieties (Co)homology theory Singularities

Published online by Cambridge University Press:  17 May 2018

Mircea Mustaţă ,
Sebastián Olano  and
Mihnea Popa
Mircea Mustaţă
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (mmustata@umich.edu)
Sebastián Olano
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA (seolano@math.northwestern.edu; mpopa@math.northwestern.edu)
Mihnea Popa
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA (seolano@math.northwestern.edu; mpopa@math.northwestern.edu)
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Abstract

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then

$$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$
We prove this when $Z$ has isolated singularities and when it is a toric variety. We deduce that for a divisor $D$ with isolated rational singularities on a smooth complex $n$-dimensional variety $X$, the generation level of Saito’s Hodge filtration on the localization $\mathscr{O}_{X}(\ast D)$ is at most $n-3$.

Keywords

rational singularitiesvanishing theoremsHodge ideals

MSC classification

Primary: 14J17: Singularities 14F17: Vanishing theorems 32S25: Surface and hypersurface singularities 32S35: Mixed Hodge theory of singular varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 801 - 819
Copyright
© Cambridge University Press 2018

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Footnotes

MM was partially supported by NSF grant DMS-1401227; MP was partially supported by NSF grant DMS-1405516.

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