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15 April 2004 Hodge cohomology of gravitational instantons
Tamás Hausel, Eugenie Hunsicker, Rafe Mazzeo
Duke Math. J. 122(3): 485-548 (15 April 2004). DOI: 10.1215/S0012-7094-04-12233-X


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 Poincaré metrics on $\mathbb{Q}$-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 $L^2$ 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.


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Tamás Hausel. Eugenie Hunsicker. Rafe Mazzeo. "Hodge cohomology of gravitational instantons." Duke Math. J. 122 (3) 485 - 548, 15 April 2004.


Published: 15 April 2004
First available in Project Euclid: 22 April 2004

zbMATH: 1062.58002
MathSciNet: MR2057017
Digital Object Identifier: 10.1215/S0012-7094-04-12233-X

Primary: 35S35, 58A14
Secondary: 35A27, 35J70

Rights: Copyright © 2004 Duke University Press


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Vol.122 • No. 3 • 15 April 2004
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