Comparison between two complexes on a singular space

Ursula Ludwig

doi:10.1515/crelle-2014-0075

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Comparison between two complexes on a singular space

Published/Copyright: September 20, 2014

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2017 Issue 724

Abstract

In this article we generalise the Witten deformation for stratified spaces and a class of Morse functions which we call radial Morse functions. In the first part of the article we perform the Witten deformation on the complex of L2-forms on a (general) stratified space. A local Spectral Gap Theorem for the model Witten Laplacian near critical points of the Morse function is proved for general stratified spaces. The global Spectral Gap Theorem for the Witten Laplacian can be generalised to spaces with isolated singularities as well as to Witt spaces. In the second part of the article we give, in the case of isolated singularities, a geometric interpretation of the complex of eigenforms of the Witten Laplacian to small eigenvalues. For this, we have to establish first an analogue of the Morse–Thom–Smale complex.

A Appendix: Integration over the model forms

Smooth critical point

Let p∈Critk⁢(fsm). Recall that, in appropriate local coordinates y1,…,yn, the Morse function has the following normal form near p:

(A.1)f=f⁢(p)+12⁢(-y12-y22-…-yk2+yk+12+…+yn2).

As is well known from the smooth theory [52] a generator for the kernel of the model Witten Laplacian ker⁡𝚫tp is given by the k-form

(A.2)ωp=(tπ)n/4⁢e-t⁢r2/2⁢d⁢y1∧…∧d⁢yk,

where r2=∑i=1nyi2. Note that ∥ωp∥=1 and

(A.3)∫Wu⁢(p)¯et⁢(f-f⁢(p))⁢ωp=(πt)k/2-n/4.

Singular point

Let p∈Sing⁢(X). The Morse function has the following normal form near the singularity: f=f⁢(p)-r2/2. Recall that ker⁡𝚫tp,(k)=0 for k<n/2+1. Moreover for k≥n/2+1 a basis of the kernel of the model Witten Laplacian 𝚫tp,(k) is given by

(A.4)φp,lk=e-t⁢r2/2⁢r2⁢(k-1)-m⁢ηp,lk-1⁢d⁢r,

where {ηp,lk-1} is an orthonormal basis of ℋk-1⁢(Lp). We have

(A.5)∥φp,lk∥2=∫e-t⁢r2⁢r4⁢(k-1)-2⁢m⁢rm-2⁢(k-1)⁢ηp,lk-1∧*~⁢ηp,lk-1⁢d⁢r

=∫e-t⁢r2⁢r2⁢(k-1)-m⁢𝑑r

=1vol⁢(S2⁢(k-1)-m)⁢(tπ)-k+n/2,

and

∫c⁢τp,rk-1et⁢(f-f⁢(p))⁢φp,lk=1vol⁢(S2⁢(k-1)-m)(tπ)-k+n/2∫c⁢τp,rk-1ηlk-1=1vol⁢(S2⁢(k-1)-m)⁢(tπ)-k+n/2⁢δr⁢l.

We define the model forms by normalising:

(A.6)ωp,lk:=vol⁢(S2⁢(k-1)-m)⁢(tπ)k/2-n/4⋅φp,lk.

Then

(A.7)∫c⁢τp,rk-1et⁢(f-f⁢(p))⁢ωp,lk=1vol⁢(S2⁢(k-1)-m)⁢(tπ)-k/2+n/4⁢∫τp,rk-1ηp,lk-1

=1vol⁢(S2⁢(k-1)-m)⁢(tπ)-k/2+n/4⁢δr⁢l.

Recall that

(A.8)vol⁢(Sk-1)=2⋅πk/2Γ⁢(k/2)={(2⁢π)k/22⋅4⁢⋯⁢(k-2),k⁢ even,2⁢(2⁢π)(k-1)/21⋅3⁢⋯⁢(k-2)=2⁢(2⁢π)(k-1)/2(k-2)!!,k⁢ odd.

Acknowledgements

I wish to thank Jean-Michel Bismut for suggesting work on the Witten deformation for singular spaces, for many discussions and support during this whole project. I am grateful to Jean-Paul Brasselet for many helpful conversations and support. I would like to thank the referee for his suggestions and questions, helping to improve the original manuscript. During the final corrections of the manuscript I had a long conversation with Eric Leichtnam as well as with Markus Banagl, and I wish to thank them for their generosity. Also during final corrections I had the opportunity for discussion with Pierre Albin, Francesco Bei, Jochen Brüning and Paolo Piazza, for which I am very grateful. During the final corrections of the manuscript the author has been supported by the Marie Curie Intra European Fellowship (within the 7th European Community Framework Programme) COMPTORSING and wishes to thank the Département de Mathématiques, Université Paris-Orsay, for hospitality.

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Received: 2011-7-19

Revised: 2014-4-26

Published Online: 2014-9-20

Published in Print: 2017-3-1

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