Witten's perturbation on strata with general adapted metrics

TL;DR

This paper extends Witten's perturbation to stratified spaces with general adapted metrics, establishing spectral discreteness, Morse inequalities, and connections to intersection homology, thus advancing analysis on singular spaces.

Contribution

It introduces a new class of adapted metrics called good metrics, proves spectral discreteness and Morse inequalities in this setting, and links cohomology to intersection homology via generalized Witten perturbation.

Findings

Discrete spectrum satisfying Weyl's law

Morse inequalities for stratified spaces

Isomorphisms with intersection homology

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/minimun ideal boundary condition, $d_{\text{\rm max/min}}$, of the compactly supported de~Rham complex on $M$, in the sense of Br\"uning-Lesch. Let $H^*_{\text{\rm max/min}}(M)$ and $\Delta_{\text{\rm max/min}}$ denote the cohomology and Laplacian of $d_{\text{\rm max/min}}$. The first main theorem states that $\Delta_{\text{\rm max/min}}$ 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 $H_{\text{\rm max/min}}^*(M)$ and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for $d_{\text{\rm max/min}}$ 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. 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 $H^r_{\text{\rm max}}(M)\cong I^{\bar p}H_r(A)^*$ ($r\in\N$). If $M$ is oriented and $\bar p\ge\bar n$, we also get $H^r_{\text{\rm min}}(M)\cong I^{\bar p}H_r(A)$. Thus our version of the Morse inequalities can be described in terms of $I^{\bar p}H_*(A)$.