Extended Hodge Theory for Fibred Cusp Manifolds

TL;DR

This paper extends Hodge theory to fibred cusp manifolds, representing intersection cohomology via weighted harmonic forms, and explores boundary value structures similar to manifolds with cylindrical ends.

Contribution

It introduces a novel representation of intersection cohomology for fibred cusp manifolds using extended weighted harmonic forms, generalizing classical Hodge theory.

Findings

Intersection cohomology groups are represented by extended weighted harmonic forms.

Boundary values of these forms define a Lagrangian splitting in the long exact sequence.

The results generalize Hodge theory to a class of pseudo manifolds with fibred cusp structures.

Abstract

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted $L^2$ harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted $L^2$ harmonic forms are harmonic forms that are almost in the given weighted $L^2$ space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. As in that setting, in the unweighted $L^2$ case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.