The Witten complex and the degenerate Morse inequalities
type: Item
instance of: scholarly article
title: The Witten complex and the degenerate Morse inequalities
zbMATH Open document ID: 0608.58038
author: Jean-Michel Bismut
published in: Journal of Differential Geometry
publication date: 1986-01-01T00:00:00Z
review text: \textit{E. Witten} [ibid. 17, 661-692 (1982; Zbl 0499.53056)] presented an analytic proof of the nondegenerate Morse inequalities, and suggested a method of proof of the degenerate Morse inequalities of Bott when the function has non-degenerate critical submanifolds. In this paper, the author uses the complex introduced by Witten, and gives a rigorous probabilistic proof of these inequalities. The method is based on an extension of the heat equation method for the proof of the index theorem by the author [J. Funct. Anal. 57, 56-99 (1984; Zbl 0538.58033)]. However, in the proof of the Morse inequalities of Bott, the Mayer- Vietoris argument is used. This question has recently received a complete analytical solution by \textit{B. Helffer} and \textit{J. Sjöstrand} [Commun. Partial Differ. Equations 10, 245-340 (1985; Zbl 0597.35024)] and 'A proof of the Bott inequalities', to appear in a volume in the honor of M. Sato.
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 3984180
zbMATH Keywords: de Rham complex
zbMATH Keywords: Witten complex
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Morse inequalities
zbMATH Keywords: critical submanifolds
zbMATH Keywords: index theorem
Wikidata QID: Q115185463
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214440113
OpenAlex ID: W1579172684
DOI: 10.4310/JDG/1214440113
description: scientific article; zbMATH DE number 3984180