Comptes Rendus
no. 9-10
Differential Geometry
Comparison between two complexes on a singular space
[Comparaison entre deux complexes sur un espace singulier]

Présenté par : Jean-Michel Bismut

Ursula Ludwig1
1 Mathematisches Institut, Eckerstrasse 1, 79104 Freiburg, Germany
Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 525-528.

Cette Note a deux buts : Dans une première partie on étend la déformation de Witten au cas dʼun espace stratifié X muni de fonctions appelées fonctions de Morse radiales. On démontre le théorème du trou spectral pour le laplacien de Witten. Dans la deuxième partie, on se place dans la situation dʼun espace à singularités isolées et on construit un complexe géométrique que lʼon compare à celui des petites valeurs propres.

The aim of this Note is twofold. In the first step we study the Witten deformation for stratified spaces X and radial Morse functions on them and prove a spectral gap theorem for the Witten Laplacian. In the second step we focus on spaces with isolated conic singularities, where we construct a geometric complex associated to the Morse function and give two comparison results.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.05.014
Ursula Ludwig 1

1 Mathematisches Institut, Eckerstrasse 1, 79104 Freiburg, Germany 
@article{CRMATH_2012__350_9-10_525_0,
     author = {Ursula Ludwig},
     title = {Comparison between two complexes on a singular space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {525--528},
     publisher = {Elsevier},
     volume = {350},
     number = {9-10},
     year = {2012},
     doi = {10.1016/j.crma.2012.05.014},
     language = {en},
}
TY  - JOUR
AU  - Ursula Ludwig
TI  - Comparison between two complexes on a singular space
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 525
EP  - 528
VL  - 350
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crma.2012.05.014
LA  - en
ID  - CRMATH_2012__350_9-10_525_0
ER  - 
%0 Journal Article
%A Ursula Ludwig
%T Comparison between two complexes on a singular space
%J Comptes Rendus. Mathématique
%D 2012
%P 525-528
%V 350
%N 9-10
%I Elsevier
%R 10.1016/j.crma.2012.05.014
%G en
%F CRMATH_2012__350_9-10_525_0
Ursula Ludwig. Comparison between two complexes on a singular space. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 525-528. doi : 10.1016/j.crma.2012.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.014/

[1] J.-M. Bismut; W. Zhang Milnor and Ray–Singer metrics on the equivariant determinant of a flat, Geom. Funct. Anal., Volume 4 (1994) no. 2, pp. 136-212

[2] J.-P. Brasselet; A. Legrand A complex of differential forms with bounded growth on a stratified manifold, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser., Volume 21 (1994) no. 2, pp. 213-234

[3] J. Brüning; M. Lesch Hilbert complexes, J. Funct. Anal., Volume 108 (1992) no. 1, pp. 88-132

[4] J. Cheeger On the spectral geometry of spaces with cone-like singularities, Proc. Natl. Acad. Sci. USA, Volume 76 (1979), pp. 2103-2106

[5] J. Cheeger On the Hodge theory of Riemannian pseudomanifolds, Honolulu, Hawaii, 1979 (Proc. Sympos. Pure Math.), Volume vol. XXXVI, Amer. Math. Soc., Providence, RI (1980), pp. 91-146

[6] M. Goresky; R. MacPherson Intersection homology theory, Topology, Volume 19 (1980), pp. 135-165

[7] M. Goresky; R. MacPherson Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3), vol. 14, Springer-Verlag, Berlin, 1988

[8] U. Ludwig, The Witten deformation for even dimensional spaces with conformally conic singularities, Trans. Am. Math. Soc., in press; announced in C. R. Acad. Sci. Paris, Ser. I 348 (15–16) (2010) 915–918.

[9] M. Nagase L2-cohomology and intersection homology of stratified spaces, Duke Math. J., Volume 50 (1983), pp. 329-368

[10] E. Witten Supersymmetry and Morse theory, J. Differential Geom., Volume 17 (1982) no. 4, pp. 661-692

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

An extension of a theorem by Cheeger and Müller to spaces with isolated conical singularities

Ursula Ludwig

C. R. Math (2018)

The Witten complex for algebraic curves with cone-like singularities

Ursula Ludwig

C. R. Math (2009)

The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions

Ursula Ludwig

C. R. Math (2010)