# Hodge filtration, minimal exponent, and local vanishing

## TL;DR

This paper establishes bounds on the Hodge filtration's generation level using the minimal exponent, leading to local vanishing theorems for sheaves with log poles, extending to Q-divisors.

## Contribution

It introduces new bounds on the Hodge filtration based on the minimal exponent and extends local vanishing results to Q-divisors.

## Key findings

- Bound on the generation level of Hodge filtration in terms of minimal exponent
- Local vanishing theorem for sheaves of forms with log poles
- Extension of results to Q-divisors

## 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 Q-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.