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LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES

  • Mircea Mustaţă (a1), Sebastián Olano (a2) and Mihnea Popa (a2)
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 $n-3$ .

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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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