Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Abstract

Over the past fifty years, Hodge and signature theorems have been proved for various classes of noncompact and incomplete Riemannian manifolds. Two of these classes are manifolds with incomplete cylindrical ends and manifolds with cone bundle ends, that is, whose ends have the structure of a fibre bundle over a compact oriented manifold, where the fibres are cones on a second fixed compact oriented manifold. In this paper, we prove Hodge and signature theorems for a family of metrics on a manifold M with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on M . We show that the Hodge and signature theorems for this family of metrics interpolate naturally between the known Hodge and signature theorems for the extremal metrics. The Hodge theorem involves intersection cohomology groups of varying perversities on the conical pseudomanifold X that completes the cone bundle metric on M . The signature theorem involves the summands i of Dai's invariant [10] that are defined as signatures on the pages of the Leray-Serre spectral sequence of the boundary fibre bundle of M . The two theorems together allow us to interpret the i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X .

Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I

“…It is interesting therefore to see how these can be seen as opposite limiting cases in a family of examples. A similar study, resulting in a family of Hodge and signature theorems, was done for singular manifolds with metrics between finite cylinders and cone bundles off a compact set in [17].…”

Hodge theory for intersection space cohomology

“…This turns out to be related to perverse signatures, which are signatures defined for arbitrary perversities on arbitrary pseudomanifolds from the extended intersection pairing on intersection cohomology. Perverse signatures are defined in the two stratum case in [21] and more generally in [16]. Theorem 1.4.…”

Intersection homology with general perversities

“…A number of authors have considered variants of intersection homology that allow more general notions of perversity, including Beilinson, Bernstein, and Deligne [5]; MacPherson [40]; King [38]; Cappell and Shaneson [11]; Habegger and Saper [32]; the author [20,22,24]; Saralegi [48]; and Hausel, Mazzeo, and Hunsicker [34][35][36]. In many of these works, perversities are still required to satisfy at least some of the prior conditions, though completely arbitrary perversities appear as far back as 1982 in the work of Beilinson, Bernstein, and Deligne on perverse sheaves, and they occur more recently in work of the author [24] and Saralegi [48].…”

“…that integration induces a linear duality isomorphism between the cohomology ofp-perverse liftable intersection differential forms and chain-theoretic perversityt −p "relative" intersection homology with real coefficients (see [48] for precise details). Non-traditional perversities also appear in an analytic setting in the works of Hausel, Hunsicker, and Mazzeo [34][35][36], in which they demonstrate that groups of L 2 harmonic forms on a manifold with fibered boundary can be identified with cohomology spaces associated to intersection cohomology groups of varying perversities for a canonical compactification of the manifold.…”

“…There has also been much interest in the analytic study of these intersection homology signatures as they arise in L 2 -cohomology and L 2 Hodge theory and as they may relate to duality in string theory, such as through Sen's conjecture on the dimension of spaces of self-dual harmonic forms on monopole moduli spaces. Results in these areas and closely related topics include those of Müller [43]; Dai [17]; Cheeger and Dai [15]; Hausel, Hunsicker, and Mazzeo [34][35][36]; Saper [45,46]; Saper and Stern [47]; and Carron [12][13][14]; and work on analytic symmetric signatures is currently being pursued by Albin, Leichtmann, Mazzeo and Piazza. Theorem 1.4 Given a pairing of local systems m 1 : E ⊗ F → G on X − X n−1 and general perversities such thatp(Z ) +q(Z ) ≤r (Z ) for all singular strata Z , then in the bounded…”