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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