A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ludwig, Ursula

doi:10.5802/aif.2657

Cited by 4 publications

(8 citation statements)

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“…By (16), (17), (21), (23) and (24) (respectively, (26)), in the case (a) (respectively, in the case (c)), we have (36) if and only if τ < 3σ (respectively, σ < 3τ + 4).…”

Section: Remark 7 (I) Ifmentioning

confidence: 94%

Witten’s perturbation on strata with general adapted metrics

2018

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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 max/min , of the compactly supported de Rham complex on M , in the sense of Brüning-Lesch. Let H * max/min (M ) and ∆ max/min denote the cohomology and Laplacian of d max/min . The first main theorem states that ∆ 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 * max/min (M ) and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for d 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 IpH * (A) with perversityp; in particular, the lower and upper middle perversities are denoted bym andn, respectively. Then, for any perversitȳ p ≤m, there is an associated good adapted metric on M satisfying the Nagase isomorphism H r max (M ) ∼ = IpHr(A) * (r ∈ N). If M is oriented andp ≥n, we also get H r min (M ) ∼ = IpHr(A). Thus our version of the Morse inequalities can be described in terms of IpH * (A).

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“…By (16), (17), (21), (23) and (24) (respectively, (26)), in the case (a) (respectively, in the case (c)), we have (36) if and only if τ < 3σ (respectively, σ < 3τ + 4).…”

Section: Remark 7 (I) Ifmentioning

confidence: 94%

Witten’s perturbation on strata with general adapted metrics

2018

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“…We will prove (5.2) for the case where, near the singularities of X, the Morse function f can be written as the real part of a holomorphic function. The general case then follows using in addition perturbation arguments as in [11]. The proof is inspired from [9], Theorem 1.4.A.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Let us mention that the Witten deformation of a singular complex curve, which is a particular example of a complex cone, has been studied already in [13] and [11].…”

Section: Introductionmentioning

confidence: 99%

An Analytic Approach to the Stratified Morse Inequalities for Complex Cones

Ludwig

2013

Int. J. Math.

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In a previous paper the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this paper, an analytic proof of this fact is given.

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“…In a previous paper [16] (see also [15]) the Witten deformation for singular complex curves C ⊂ P n (C) equipped with a stratified Morse function in the sense of the theory in [12] has been discussed. There the local model has a particularly simple form which has been treated by an explicit computation.…”

Section: )mentioning

confidence: 99%

“…As in [15], Section 4, one can now show that the linear map P i (t, [0, 1]) • J i (t) : C i −→ S i t is a bijective map from C i onto S i t and thus the complex (S t , d t , , ) is a finite dimensional subcomplex of (C t , d t , , ) with dim S i t = c i (f ). By Proposition 2.6 moreover H * (2) (X) ≃ ker(∆ t ) ≃ H * (S t , d t , , ).…”

Section: Proof Of the Spectral Gap Theorem And Morse Inequalitiesmentioning

confidence: 99%

The Witten deformation for even dimensional conformally conic manifolds

Ludwig

2012

Trans. Amer. Math. Soc.

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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)

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A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Cited by 4 publications

References 10 publications

Witten’s perturbation on strata with general adapted metrics

Witten’s perturbation on strata with general adapted metrics

An Analytic Approach to the Stratified Morse Inequalities for Complex Cones

The Witten deformation for even dimensional conformally conic manifolds

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