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We bound the generation level of the Hodge filtration on the localizationalong a hypersurface in terms of its minimal exponent. As a consequence, weobtain a local vanishing theorem for sheaves of forms with log poles. Theseresults are extended to Q-divisors, and are derived from a result ofindependent interest on the generation level of the Hodge filtration on nearbyand vanishing cycles.

Hodge filtration, minimal exponent, and local vanishing