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 $${\mathbf {Q}}$$ Q -divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.

中文翻译：

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Hodge 过滤、最小指数和局部消失

我们根据其最小指数将 Hodge 过滤的生成级别限制在沿超曲面的定位上。因此，我们获得了具有对数极点形式的绳轮的局部消失定理。这些结果被扩展到 $${\mathbf {Q}}$$ Q -divisors，并且源自对附近和消失循环的 Hodge 过滤的生成级别的独立兴趣的结果。