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 Bruning–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 $$I^{\bar{p}}H_*(A)$$ .