Rational Homology Manifolds and Hypersurface Normalizations

TL;DR

This paper establishes a criterion to identify when the normalization of a complex analytic space is a rational homology manifold, utilizing a specific perverse sheaf linked to parameterized spaces and related to Milnor monodromy and Hodge theory.

Contribution

It introduces a new criterion based on the multiple-point complex to determine rational homology manifold status of space normalizations, connecting perverse sheaves with monodromy and Hodge modules.

Findings

Provides a criterion for rational homology manifolds via perverse sheaves.

Connects the multiple-point complex with Milnor monodromy.

Links perverse sheaves to mixed Hodge Modules.

Abstract

We prove a criterion for determining whether the normalization of a complex analytic space on which the constant sheaf is perverse is a rational homology manifold, using a perverse sheaf known as the multiple-point complex. This perverse sheaf is naturally associated to any "parameterized space", and has several interesting connections with the Milnor monodromy and mixed Hodge Modules.