Weighted Hodge ideals of reduced divisors

TL;DR

This paper introduces weighted Hodge ideals derived from filtrations on hypersurface localizations, connecting birational geometry and $V$-filtration to study hypersurface singularities.

Contribution

It defines and analyzes weighted Hodge ideals, expanding the understanding of their properties and applications in hypersurface singularity theory.

Findings

Weighted Hodge ideals include adjoint and multiplier ideals.

They have specific local and global properties.

Applications to hypersurface singularities are demonstrated.

Abstract

We study the Hodge and weight filtrations on the localization along a hypersurface, using methods from birational geometry and the $V$-filtration induced by a local defining equation. These filtrations give rise to ideal sheaves called weighted Hodge ideals, which include the adjoint ideal and a multiplier ideal. We analyze their local and global properties, from which we deduce applications related to singularities of hypersurfaces of smooth varieties.