Hodge cohomology of gravitational instantons

Abstract

We study the space of L2 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 L2 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 -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 L2 harmonic forms in duality theories in string theory.