Hodge theorem for iterated fibred cusp metrics on non-Witt stratified spaces of arbitrary depth

For a smoothly stratified space $X$ that is not Witt, the unweighted $L^2$ cohomology of a quasi iterated fibred cusp metric on the regular stratum is typically infinite-dimensional, so the finite-dimensional Hodge isomorphism between $L^2$ cohomology, $L^2$ harmonic forms, and middle-perversity intersection cohomology (proved by Hunsicker and Rochon in the Witt case) breaks down. The problem is to prove a Hodge theorem in this non-Witt, arbitrary-depth setting: build the analytic machinery (self-adjoint/Fredholm realization of the Hodge–de Rham operator, decay of $L^2$ harmonic forms at the strata) sufficient to control the space of $L^2$ harmonic forms, and identify it naturally with a topological invariant of $X$ — expected, by analogy with the depth-one theorem of Hausel–Hunsicker–Mazzeo, to be the image of lower-middle-perversity in upper-middle-perversity intersection cohomology. The extension beyond one singular stratum, posed by Hunsicker and Rochon in 2015, remains open: later work treats the non-Witt case only for incomplete edge metrics with mezzoperversity boundary conditions, for conformally related quasi-fibered cusp metrics under hypotheses that force the Witt condition, or for alternative invariants on spaces with a single singular stratum.

Background

Cheeger discovered that the classical Hodge theorem survives on a class of singular spaces: if $X$ is a compact stratified pseudomanifold endowed with an (incomplete) iterated conical metric on its regular part, the $L^2$ cohomology computes intersection cohomology; Cheeger–Goresky–MacPherson developed the corresponding $L^2$-cohomology theory for singular algebraic varieties. Recall the ingredients. A smoothly stratified space $X$ is a pseudomanifold partitioned into smooth manifolds (strata) such that each singular stratum $S$ has a neighborhood modeled on a cone over a link $L$ crossed with a ball, where $L$ is itself a smoothly stratified space of lower depth; the depth of $X$ is the maximal nesting of strata. Intersection cohomology $IH^*_p(X)$, introduced by Goresky and MacPherson, is a topological invariant depending on a perversity $p$, a function of the codimension $c$ of a stratum controlling how chains may meet it; the lower middle and upper middle perversities are $\bar m(c)=\lfloor (c-2)/2\rfloor$ and $\bar n(c)=\lceil (c-2)/2\rceil$. The space $X$ is called Witt if $IH^{\dim L/2}_{\bar m}(L)=\{0\}$ for every even-dimensional link $L$; exactly under this condition the lower- and upper-middle theories coincide and satisfy Poincaré duality, giving a unique middle-perversity intersection cohomology. For a complete Riemannian metric $g$ on the regular part $M=X\setminus X_{\mathrm{sing}}$, the $L^2$ cohomology is the cohomology of the complex of square-integrable forms whose exterior derivative is also square-integrable, and $\mathcal H^*_{L^2}(M,g)$ denotes the space of $L^2$ forms annihilated by the Hodge–de Rham operator $D=d+\delta$, where $\delta$ is the formal adjoint of $d$.

Hunsicker and Rochon replaced the incomplete conical metrics by a natural class of complete metrics. Resolving $X$ by a manifold with fibred corners (in the framework of Debord–Lescure–Rochon), a quasi iterated fibred cusp metric on $M$ is a complete metric quasi-isometric to an iterated fibred cusp metric; near a singular stratum it has the model form

$$g_U=\frac{dx^2}{x^2}+g_V+x^2 g_L,$$

where $g_L$ is a quasi iterated fibred cusp metric on the interior of the link, i.e. generalized hyperbolic cusp behavior in the cone directions. Their Theorem 2 identifies, for any perversity $p$, the weighted $L^2$ cohomology with respect to the weight $w_p=x_1^{i_1}\cdots x_n^{i_n}$ (with $i_j=p(c_j)-(c_j-3)/2+\epsilon$, $0<\epsilon<1/2$, for boundary hypersurfaces of codimension $c_j$) with the perversity-$p$ intersection cohomology $IH^*_p(X)$, and by the Kodaira decomposition of Hilbert complexes also with the space of weighted $L^2$ harmonic forms. In the Witt case (Theorem 4) the weight can be dropped: the unweighted $L^2$ cohomology, the space of $L^2$ harmonic forms, and the middle-perversity intersection cohomology $IH^*_{\bar m}(X)$ are naturally isomorphic, and the signature operator is Fredholm with index the intersection-cohomology signature.

The obstruction in the non-Witt case is that the unweighted $L^2$ cohomology is typically infinite-dimensional (the weighted computation produces an infinite-dimensional summand $WH^1((0,1),dx^2/x^2,a)$ exactly when the weight exponent $a$ vanishes, which the Witt condition rules out), so the finite-dimensional Hilbert-complex argument behind Theorem 4 collapses. In the single-stratum (depth-one) case the corresponding harder analysis was carried out by Hausel–Hunsicker–Mazzeo: using the Mazzeo–Melrose pseudodifferential calculus for fibred boundary operators (with related edge analysis in Hunsicker–Mazzeo), they proved that the space of $L^2$ harmonic forms for a manifold with fibred boundary endowed with a fibred cusp metric is nevertheless finite-dimensional and naturally isomorphic to the image of lower-middle-perversity in upper-middle-perversity intersection cohomology of its fibrewise compactification; Gell-Redman–Rochon generalized this to some foliated boundary and foliated cusp metrics. Hunsicker–Rochon state that the extension to iterated fibred cusp metrics on non-Witt spaces of arbitrary depth requires harder analytic results of this type and leaves the problem open.

Several adjacent programs address the non-Witt analytic difficulty by other routes, none of which answers the question for complete iterated fibred cusp metrics. Albin–Leichtnam–Mazzeo–Piazza developed the ideal-boundary-condition approach: on Cheeger spaces (stratified spaces admitting a self-dual mezzoperversity, a choice interpolating between the lower and upper middle perversities), the de Rham operator with ideal boundary conditions is Fredholm and its Hodge and $L^2$ de Rham cohomology satisfy Poincaré duality — but for incomplete iterated edge metrics and with modified domains, not the raw $L^2$ harmonic forms of a complete metric; their companion work with Banagl classifies the constructible sheaf complexes (refined intersection homology) relevant to the non-Witt perverse-sheaf picture. Banagl–Hunsicker proved a Hodge theorem for non-Witt two-strata spaces realizing Banagl's intersection space cohomology by $L^2$ cohomology of fibered scattering metrics. Most directly, Kottke–Rochon studied quasi-fibered boundary and the conformally related quasi-fibered cusp (QFC) metrics — a different generalization of fibred cusp metrics to higher depth without warped-product local models — computing their weighted $L^2$ cohomology for arbitrary perversities and, via a new pseudodifferential calculus, the decay of $L^2$ harmonic forms; their unweighted Hodge identification, however, holds under hypotheses (all links odd-dimensional with vanishing middle-degree lower-middle intersection cohomology) that force the space to be Witt. The non-Witt, arbitrary-depth, unweighted question for iterated fibred cusp metrics thus remains unresolved.

Problem Statement

Let $X$ be a compact smoothly stratified space of arbitrary depth that is not Witt, and let $g$ be a quasi iterated fibred cusp metric on the regular stratum $M=X\setminus X_{\mathrm{sing}}$, that is, a complete metric quasi-isometric to an iterated fibred cusp metric, locally near each singular stratum of the model form $dx^2/x^2+g_V+x^2 g_L$ with $g_L$ a quasi iterated fibred cusp metric on the link. Prove a Hodge theorem for $(M,g)$ in this non-Witt setting. Concretely: (1) establish the analytic foundation — essential self-adjointness, or a natural self-adjoint and Fredholm realization, of the Hodge–de Rham operator $D=d+\delta$ on $L^2$ forms over $(M,g)$, closed range of $d$ and its adjoint on the relevant domains, and sufficient decay (e.g. polyhomogeneous asymptotics, meaning expansions in powers of the boundary defining functions and their logarithms) of $L^2$ harmonic forms at the singular strata — that replaces the finite-dimensional Hilbert-complex argument which fails because the unweighted $L^2$ cohomology is typically infinite-dimensional; and (2) identify the space $\mathcal H^*_{L^2}(M,g)$ of $L^2$ harmonic forms (and, where it is finite-dimensional, the reduced $L^2$ cohomology) naturally, and independently of the metric within its quasi-isometry class, with a topological invariant of the stratification. The expected invariant, by analogy with the depth-one theorem of Hausel–Hunsicker–Mazzeo, is the image $\operatorname{Im}\big(IH^*_{\bar m}(X)\to IH^*_{\bar n}(X)\big)$ of lower-middle-perversity in upper-middle-perversity intersection cohomology; the alternative invariant is of mezzoperversity or refined intersection cohomology type, and any such identification must specialize on Witt spaces to $IH^*_{\bar m}(X)$ (Hunsicker–Rochon, Theorem 4) and on depth-one spaces to the Hausel–Hunsicker–Mazzeo theorem. If the expected identification fails for some non-Witt stratified space of depth at least two with a quasi iterated fibred cusp metric, the question is what the harmonic forms compute instead.

References

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