Abstract
We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are extended to \({\mathbf {Q}}\)-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.
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Notes
As mentioned above, for \(n=2\) the filtration is always generated at level 0.
Note that D has rational singularities if and only if \({\widetilde{\alpha }}_D > 1\), so the case \(i = n-2\) corresponds to the statements in loc. cit.
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Acknowledgements
We thank the referee for very useful comments that helped us improve the exposition.
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MM was partially supported by NSF Grant DMS-1701622 and a Simons Fellowship; MP was partially supported by NSF Grant DMS-1700819.
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Mustaţă, M., Popa, M. Hodge filtration, minimal exponent, and local vanishing. Invent. math. 220, 453–478 (2020). https://doi.org/10.1007/s00222-019-00933-x
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DOI: https://doi.org/10.1007/s00222-019-00933-x