posted on 2009-03-04, 10:18authored byTamás Hausel, Eugenie Hunsicker, Rafe Mazzeo
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.
History
School
Science
Department
Mathematical Sciences
Citation
HAUSEL, T., HUNSICKER, E. and MAZZEO, R., 2004. Hodge cohomology of gravitational instantons. Duke Mathematical Journal, 122 (3), pp. 485-548