On the Hodge Theory of Stratified Spaces

ON THE HODGE THEORY OF STRATIFIED SPACES

PIERRE ALBIN

Dedicated to Steven Zucker on the occasion of his 65th birthday

Abstract. This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of a perversity in the sense of Goresky-MacPherson to a mezzoperversity. We define Cheeger spaces and their signatures and describe how to carry out the analytic proof of the Novikov conjecture on these spaces. Finally we review the reductive Borel-Serre compactification of a locally symmetric space to a stratified space and describe its resolution to a manifold with corners.

1. Introduction The purpose of this paper is to describe some recent advances in the Hodge theory of pseudomanifolds.

Algebraic varieties and L2-metrics have been integral aspects of Hodge theory from the very beginning. Hodge appealed to the Dirichlet principle [60] and the Levi parametrix method [61, 62] to find harmonic forms in each de Rham cohomology class and used these forms to study the cohomology of complex projective algebraic varieties. Weyl [116] referred to [62] as “one of the great landmarks in the history of our science in the present century” (and corrected Hodge’s existence proof in [115]). An existence proof using purely Hilbert space methods was carried out by Gaffney [47, 48] who, for instance, considered the exterior derivative d on a smooth manifold X as an unbounded operator on square integrable differential forms with domain

2 • ∗ 2 • ∗ Dmax(d) = {ω ∈ L (X;Λ T X): dω ∈ L (X;Λ T X)}, where dω is computed distributionally. This domain forms a complex whose cohomology, known as the L2-cohomology of X, was introduced independently in [118] on a complete manifold and in [36] on an incomplete manifold. Algebraic varieties are often singular but Whitney [67,117] showed that they are stratified spaces. This means that they can be decomposed into smooth manifold pieces of various dimensions, called strata, in such a way that different points on the same stratum have sim- ilar neighborhoods in the variety. In particular one of these pieces is an open dense smooth manifold. Endowing this ‘regular part’ with a conic-type Riemannian metric, Cheeger stud- ied the de Rham complex of differential forms in L2. He singled out a topological condition that guaranteed that the exterior derivative had an unambiguous definition as an unbounded operator, every cohomology class had a harmonic representative, and the resulting cohomology groups satisfy Poincaréduality. In the case of conic singularities he explained how, if this topological condition failed, one can choose ideal boundary conditions and still establish 1 2 PIERRE ALBIN

these results. In this note I will review recent work [1, 3–5] extending Cheeger’s results to general stratified ‘pseudomanifolds’. Specifically, let Xb denote a pseudomanifold (a stratified space ‘without boundary’) and endow its regular part Xbreg with a ‘wedge metric’ g. The exterior derivative d and its formal adjoint δ make up the de Rham operator ∞ reg • ∗ reg ∞ reg • ∗ reg ðdR = d + δ : Cc (Xb ;Λ T Xb ) −→ Cc (Xb ;Λ T Xb ). 2 reg • ∗ reg There are two canonical extensions of ðdR to a closed operator on L (Xb ;Λ T Xb ). The first has domain 2 reg • ∗ reg 2 reg • ∗ reg Dmax(ðdR) = {ω ∈ L (Xb ;Λ T Xb ): ðdRω ∈ L (Xb ;Λ T Xb )} where ðdRω is computed distributionally, and the second has domain

2 reg • ∗ reg ∞ reg • ∗ reg 2 Dmin(ðdR) = {ω ∈ L (Xb ;Λ T Xb ): ∃(ωn) ⊆ Cc (Xb ;Λ T Xb ) s.t. ωn → ω in L 2 and (ðdRωn) is L -Cauchy} and ðdRω can be computed both distributionally and as lim ðdRωn. Cheeger [36] singled out a class of spaces, now known as Witt