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Hodge cohomology of gravitational instantons. (English) Zbl 1062.58002

For a compact Riemannian manifold \((M,g)\) the Hodge theorem identifies the space \(L^ 2{\mathcal H}^*(M,g)\) of \(L^ 2\) harmonic forms on \(M\) with the de Rham cohomology of this space. If \(M\) is not compact there is no general theorem that identifies the space \(L^ 2{\mathcal H}^*(M,g)\) with a topologically defined group. However, there are topological interpretations of this Hodge cohomology space.
In this paper, the authors prove the Hodge-type theorem for two different classes of Riemannian manifolds, manifolds with fibered boundary and fibered cusp metrics. A metric \(g\) on \(M\) is said to be of fibered boundary type if there is a compactification of \(M\) to a manifold with boundary \(\overline{M}\), with a boundary-defining function \(x\), and a fibration \(\varphi : \partial M \to B\), such that in local coordinates near the boundary, \(g\) takes the form \(g = \frac{dx^ 2}{x^ 4} + \frac{\tilde h}{x^ 2}+k\), where \(h\) is a smooth metric on \(B\), \(\tilde h\) is a smooth extension of \(\varphi^* h\) from \(\partial M\) to \(\overline{M}\), and \(k\) is a smooth symmetric 2-tensor on \(\overline{M}\) which is positive definite on each fiber of \(\partial M\). A metric \(g\) is a fibered cusp metric if it is of the form \(x^ 2 g\), where \(g\) is a fibered boundary metric.
Let \(X\) be the stratified space defined as the quotient space of \(\overline{M}\) with the fibers on the boundary of \(\overline{M}\) identified to points. Thus \(X\) has principal stratum \(M\) and singular stratum \(B\). If the metric \(g\) on \(M\) is of fibered boundary type and \(b=\dim B\) is even, then the authors prove that the space \(L^ 2{\mathcal H}^*(M,g)\) of \(L^ 2\) harmonic forms \(k\)-forms is naturally isomorphic to \(IH_{f+b/2-k}^ k(X, B)\), where \(f\) is the dimension of the fibers and the space \(IH_j^k(X, B)\) is defined as \(H^k(M)\) when \(j \leq -1\), as \(H^k(\overline{M}, \partial M)\) when \(j \geq f\), and as \(IH^k_p(X)\) otherwise, where \(p\) is any perversity with \(p(f+1)=j\). Similar theorems are proven for the case when \(b\) is odd and for the case when the metric \(g\) is of fibered cusp type.

MSC:

58A14 Hodge theory in global analysis
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
35J70 Degenerate elliptic equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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