Hausel, TamasIST Austria; Hunsicker, Eugénie; Mazzeo, Rafe R
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 Poincaré metrics on ℚ-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kähler 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 and more generally by Vaillant and identifies Dai's τ-invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron and the forthcoming paper of Cheeger and Dai. 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.
Duke Mathematical Journal
Hausel’s work supported by a Miller Research Fellowship at the University of California, Berkeley. Hunsicker’s work partially supported by Stanford University. Mazzeo’s work supported by National Science Foundation grant numbers DMS-991975 and DMS-0204730 and by the Mathematical Sciences Research Institute.
485 - 548
Hausel T, Hunsicker E, Mazzeo R. Hodge cohomology of gravitational instantons. Duke Mathematical Journal. 2004;122(3):485-548. doi:10.1215/S0012-7094-04-12233-X
Hausel, T., Hunsicker, E., & Mazzeo, R. (2004). Hodge cohomology of gravitational instantons. Duke Mathematical Journal. Duke University Press. https://doi.org/10.1215/S0012-7094-04-12233-X
Hausel, Tamás, Eugénie Hunsicker, and Rafe Mazzeo. “Hodge Cohomology of Gravitational Instantons.” Duke Mathematical Journal. Duke University Press, 2004. https://doi.org/10.1215/S0012-7094-04-12233-X.
T. Hausel, E. Hunsicker, and R. Mazzeo, “Hodge cohomology of gravitational instantons,” Duke Mathematical Journal, vol. 122, no. 3. Duke University Press, pp. 485–548, 2004.
Hausel T, Hunsicker E, Mazzeo R. 2004. Hodge cohomology of gravitational instantons. Duke Mathematical Journal. 122(3), 485–548.
Hausel, Tamás, et al. “Hodge Cohomology of Gravitational Instantons.” Duke Mathematical Journal, vol. 122, no. 3, Duke University Press, 2004, pp. 485–548, doi:10.1215/S0012-7094-04-12233-X.