Hodge and signature theorems for a family of manifolds with fibration boundary

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Let M be a manifold with boundary Y which is the total space of a fibre bundle, and is defined by the vanishing of a boundary defining function, x. We prove L² Hodge and signature theorems for M endowed with a metric of the form dx² + x²ᶜ h + k, where k is the lift to Y of the metric on the base of the fibre bundle, h is a two form on Y which restricts to a metric on each fibre, and 0 ≤ c ≤ 1. These metrics interpolate between the case when c=0, in which case the metric near the boundary is a cylinder, and the case where c=1, in which case the metric near the boundary is that of a cone bundle over the base of the boundary fibration. We show that the L² Hodge theorems for the cohomologies given by the maximal and minimal extensions of d with respect to these metrics and the L² signature theorem for the image of the minimal cohomology in the maximal cohomology interpolate between known results for L² Hodge and signature theorems for cylindrical and cone bundle type metrics. In particular, the Hodge theorems all relate the related spaces of L² harmonic forms to intersection cohomology of varying perversities for X, the space formed from M by collapsing the fibres of Y at the boundary. The signature theorem involves variations on the τ invariant described by Dai.