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LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 3 / May 2020
- Published online by Cambridge University Press:
- 17 May 2018, pp. 801-819
- Print publication:
- May 2020
-
- Article
- Export citation
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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 We prove this when
$$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$
$Z$ 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 
$n-3$.
