Duke Mathematical Journal

Hodge cohomology of gravitational instantons

Tamás Hausel, Eugenie Hunsicker, and Rafe Mazzeo

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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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Duke Math. J., Volume 122, Number 3 (2004), 485-548.

First available in Project Euclid: 22 April 2004

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Primary: 58A14: Hodge theory [See also 14C30, 14Fxx, 32J25, 32S35] 35S35: Topological aspects: intersection cohomology, stratified sets, etc. [See also 32C38, 32S40, 32S60, 58J15]
Secondary: 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15] 35J70: Degenerate elliptic equations


Hausel, Tamás; Hunsicker, Eugenie; Mazzeo, Rafe. Hodge cohomology of gravitational instantons. Duke Math. J. 122 (2004), no. 3, 485--548. doi:10.1215/S0012-7094-04-12233-X. https://projecteuclid.org/euclid.dmj/1082665286

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