Witten’s perturbation on strata with general adapted metrics

Abstract

Abstract. Let M be a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then g is called good. We consider the maximum/minimum ideal boundary condition, d max/min , of the compactly supported de Rham complex on M , in the sense of Brüning-Lesch. Let H * max/min (M ) and ∆ max/min denote the cohomology and Laplacian of d max/min . The first main theorem states that ∆ max/min has a discrete spectrum satisfying a weak form of the Weyl's asymptotic formula. The second main theorem is a version of Morse inequalities using H * max/min (M ) and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for d max/min of the Witten's perturbation of the de Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory.The condition on g to be good is general enough in the following sense. Assume that A is a stratified pseudomanifold, and consider its intersection homology IpH * (A) with perversityp; in particular, the lower and upper middle perversities are denoted bym andn, respectively. Then, for any perversitȳ p ≤m, there is an associated good adapted metric on M satisfying the Nagase isomorphism H r max (M ) ∼ = IpHr(A) * (r ∈ N). If M is oriented andp ≥n, we also get H r min (M ) ∼ = IpHr(A). Thus our version of the Morse inequalities can be described in terms of IpH * (A).