Extended Hodge theory for fibred cusp manifolds

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 [Formula: see text] harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted [Formula: see text] harmonic forms are harmonic forms that are almost in the given weighted [Formula: see text] 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. In analogy with that setting, in the unweighted [Formula: see text] 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.