Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Abstract

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 $M$ with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on $M$. 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 $X$ that completes the cone bundle metric on $M$. The signature theorem involves the summands ${\tau }_i$ of Dai’s $\tau$ 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 $M$. The two theorems together allow us to interpret the ${\tau }_i$ in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on $X$.