RT Journal Article
T1 Witten’s perturbation on strata with general adapted metrics
A1 Álvarez López, Jesús Antonio
A1 Calaza Cabanas, Manuel
A1 Franco, Carlos
K1 Morse inequalities
K1 Witten's perturbation
K1 Ideal boundary condition
K1 Stratification
K1 General adapted metric
AB 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, $d_{\text{\rm max/min}}$, 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.      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 $H^r_{\text{\rm max}}(M)\cong I^{\bar p}H_r(A)^*$ ($r\in\N$). 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 $I^{\bar p}H_*(A)$.
PB Springer
YR 2018
FD 2018-01-13
LK http://hdl.handle.net/10347/32034
UL http://hdl.handle.net/10347/32034
LA eng
NO Álvarez López, J.A., Calaza, M., Franco, C. (2018). Witten’s perturbation on strata with general adapted metrics. "Ann. Global Anal. Geom.", vol. 54, 25-69.
NO MICINN, Grant MTM2011-25656, and by MEC, Grant MTM2014-56950-P [A.L.]; Xunta de Galicia and the European Union (European Social Fund - ESF) [C.F.].
DS Minerva
RD 24 abr 2026