# Local vanishing and Hodge filtration for rational singularities

## TL;DR

This paper investigates the vanishing of certain higher direct images for varieties with rational singularities and explores implications for the Hodge filtration on localizations of structure sheaves.

## Contribution

It formulates a conjecture on vanishing properties and proves it for isolated singularities and toric varieties, linking to Hodge filtration generation levels.

## Key findings

- Proved vanishing conjecture for isolated singularities.
- Established vanishing for toric varieties.
- Bound the generation level of Hodge filtration to at most n-3.

## Abstract

Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. 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 of the structure sheaf along D is at most n-3.