Hodge cohomology of gravitational instantons

We study the space of L-2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincare metrics on Q-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kahler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L-2 signature formula implied by our result is closely related to the one proved by Dai [25] and more generally by Vaillant [67], and identifies Dai's tau-invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [12] and the forthcoming paper of Cheeger and Dai [17]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about L-2 harmonic forms in duality theories in string theory.