Weights on cohomology, invariants of singularities, and dual complexes

TL;DR

This paper introduces invariants derived from the Deligne weight filtration to analyze singularities, providing bounds and restrictions on their topology, and aims to make these concepts more accessible.

Contribution

It develops natural invariants of singularities using the Deligne weight filtration and establishes explicit bounds for various classes, enhancing understanding of their topology.

Findings

Bounds on weights for rational, Cohen-Macaulay, and toroidal singularities

Restrictions on the topology of singularities based on weight invariants

Explicit descriptions of the weight filtration for standard classes

Abstract

In this paper, we extract natural invariants of a singularity by using the Deligne weight filtration on the cohomology of an exceptional fibre of a resolution, and also on the intersection cohomology of the link. Our primary goal is to study and give natural bounds on the weights in terms of direct images of differential forms. These bounds can be made explicit for various standard classes such as rational, isolated normal Cohen-Macaulay and toroidal singularities, and lead to strong restrictions on the topology of these singularities. A secondary goal of this paper is to make the weight filtration, and related constructions, more widely accessible. So we have tried to make the presentation somewhat self contained. This is supersedes our earlier preprint arXiv:0902.4234.