Over the past fifty years, Hodge and signature theorems have been proved
for various classes of noncompact and incomplete Riemannian manifolds.
Two of these classes are manifolds with incomplete cylindrical ends and
manifolds with cone bundle ends, that is, whose ends have the structure
of a fibre bundle over a compact oriented manifold, where the fibres are
cones on a second fixed compact oriented manifold. In this paper, we prove
Hodge and signature theorems for a family of metrics on a manifold
with fibre bundle
boundary that interpolates between the incomplete cylindrical metric and the cone bundle
metric on .
We show that the Hodge and signature theorems for this family of
metrics interpolate naturally between the known Hodge and signature
theorems for the extremal metrics. The Hodge theorem involves intersection
cohomology groups of varying perversities on the conical pseudomanifold
that completes the
cone bundle metric on .
The signature theorem involves the summands
of
Dai’s
invariant [J Amer Math Soc 4 (1991) 265–321] that are defined as signatures on the
pages of the Leray–Serre spectral sequence of the boundary fibre bundle of
.
The two theorems together allow us to interpret the
in terms of perverse signatures, which are signatures defined on
the intersection cohomology groups of varying perversities on
.
