The Witten deformation for even dimensional conformally conic manifolds

Abstract

The goal of this article is to generalise the Witten deformation to even dimensional conformally conic manifolds X and a class of functions f : X → R called admissible Morse functions. We get Morse inequalities relating the L 2 -Betti numbers of X with the number of critical points of the function f . Hereby the contribution of a singular point p of X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse data of f at p. The definition of an admissible Morse function is inspired by stratified Morse theory as developed by Goresky and MacPherson.MSC-class 35A20 (Primary), 57R70 (Secondary)

An Analytic Approach to the Stratified Morse Inequalities for Complex Cones

“…In [12] Morse theory on singular spaces has been approached using the analytic method of the Witten deformation. The class of spaces considered in [12] are conformally conic Riemannian manifolds in the sense defined in [3]. Those are a generalisation of spaces with cone-like singularities.…”

“…Their intersection cohomology has an analytic expression in terms of the cohomology of the complex of L 2 -forms. The Witten deformation (proposed in [14], rigorously proven in [10] for the smooth situation) generalised to a conformally conic Riemannian manifold X consists in deforming the complex of L 2 -forms using a certain class of functions, which were called admissible Morse functions in [12]. One gets again Morse inequalities, this time relating the L 2 -Poincaré polynomial to the number of critical points of the Morse function.…”

“…Again the singular points of X contribute to this Morse inequalities by dim IH i (cone(L p ), l − p ) in degree i. However, in the analytic proof of the Morse inequalities in [12], the contribution of the singularities to the Morse inequalities are computed in terms of the model Witten Laplacian ∆…”

“…Note that in the general situation treated in [12] the contribution of the singularity to the Morse inequalities is not necessarily concentrated in middle degree only and in general depends on the Morse function. However if X is a complex cone and f : X → R is a stratified Morse function, then f is also admissible in the sense of [12] (see Proposition 2.1) and comparing the results in [12] and Theorem 1.1 one gets that Theorem 1.2. Let X be a complex cone and f : X → R a stratified Morse function, then…”

“…The goal of this paper is to give an analytic proof of Theorem 1.2. We use the results in [12], the Lefschetz Theorem for conformally conic Kähler manifolds (see [3]) together with a trick inspired from "Gromov's trick" in [9] (Theorem 1.4.A).…”