Witten’s perturbation on strata with general adapted metrics

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, $d_{\mathrm{max/min}}$, of the compactly supported de Rham complex on M, in the sense of Brüning–Lesch. Let $H^*_{\mathrm{max/min}}(M)$ and $\Delta _{\mathrm{max/min}}$ denote the cohomology and Laplacian of $d_{\mathrm{max/min}}$. The first main theorem states that $\Delta _{\mathrm{max/min}}$ 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 $H_{\mathrm{max/min}}^*(M)$ and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for $d_{\mathrm{max/min}}$ 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 $I^{\bar{p}}H_*(A)$ with perversity $\bar{p}$; in particular, the lower and upper middle perversities are denoted by $\bar{m}$ and $\bar{n}$, respectively. Then, for any perversity $\bar{p}\le \bar{m}$, there is an associated good adapted metric on M satisfying the Nagase isomorphism $H^r_{\mathrm{max}}(M)\cong I^{\bar{p}}H_r(A)^*$ ($r\in \mathbb {N}$). If M is oriented and $\bar{p}\ge \bar{n}$, we also get $H^r_{\mathrm{min}}(M)\cong I^{\bar{p}}H_r(A)$. Thus our version of the Morse inequalities can be described in terms of $I^{\bar{p}}H_*(A)$.