Comptes Rendus
no. 1
Differential geometry
An index formula for the intersection Euler characteristic of an infinite cone
[Un théorème de l'indice pour la caractéristique d'Euler d'intersection d'un cône infini]

Présenté par : the Editorial Board

Ursula Ludwig1
1 Universität Duisburg–Essen, Fakultät für Mathematik, 45117 Essen, Germany
Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 94-98.

Le but de cette note est d'établir un théorème de l'indice pour la caractéristique d'Euler d'intersection d'un cône. Dans un premier temps, on étudie les propriétés spectrales du laplacien de Witten et on établit une formule de McKean–Singer. On donne aussi une formule explicite pour la fonction zêta associée au laplacien de Witten. Dans une deuxième partie, on applique des techniques d'indice local au laplacien de Witten. On obtient une formule qui exprime la caractéristique d'Euler d'intersection du cône comme la somme de deux termes, l'un local, l'autre étant l'invariant de Cheeger.

The aim of this note is to establish an index formula for the intersection Euler characteristic of a cone. The main actor of these notes is the model Witten Laplacian on the infinite cone. First, we study its spectral properties and establish a McKean–Singer-type formula. We also give an explicit formula for the zeta function of the model Witten Laplacian. In a second step, we apply local index techniques to the model Witten Laplacian. By combining these two steps, we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, a term which is local, and a second term which is the Cheeger invariant.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.11.003

Ursula Ludwig 1

1 Universität Duisburg–Essen, Fakultät für Mathematik, 45117 Essen, Germany 
@article{CRMATH_2017__355_1_94_0,
     author = {Ursula Ludwig},
     title = {An index formula for the intersection {Euler} characteristic of an infinite cone},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {94--98},
     publisher = {Elsevier},
     volume = {355},
     number = {1},
     year = {2017},
     doi = {10.1016/j.crma.2016.11.003},
     language = {en},
}
TY  - JOUR
AU  - Ursula Ludwig
TI  - An index formula for the intersection Euler characteristic of an infinite cone
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 94
EP  - 98
VL  - 355
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2016.11.003
LA  - en
ID  - CRMATH_2017__355_1_94_0
ER  - 
%0 Journal Article
%A Ursula Ludwig
%T An index formula for the intersection Euler characteristic of an infinite cone
%J Comptes Rendus. Mathématique
%D 2017
%P 94-98
%V 355
%N 1
%I Elsevier
%R 10.1016/j.crma.2016.11.003
%G en
%F CRMATH_2017__355_1_94_0
Ursula Ludwig. An index formula for the intersection Euler characteristic of an infinite cone. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 94-98. doi : 10.1016/j.crma.2016.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.003/

[1] J.-M. Bismut; J. Cheeger Families index for manifolds with boundary, superconnections, and cones. I: Families of manifolds with boundary and Dirac operators, J. Funct. Anal., Volume 89 (1990) no. 2, pp. 313-363

[2] J.-M. Bismut; W. Zhang An Extension of a Theorem by Cheeger and Müller, Astérisque, vol. 205, Société mathématique de France, Paris, 1992 (with an appendix by François Laudenbach)

[3] J. Brüning; M. Lesch Kähler–Hodge theory for conformal complex cones, Geom. Funct. Anal., Volume 3 (1993) no. 5, pp. 439-473

[4] J. Brüning; R. Seeley The resolvent expansion for second order regular singular operators, J. Funct. Anal., Volume 73 (1987), pp. 369-429

[5] J. Cheeger Spectral geometry of singular Riemannian spaces, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 575-657

[6] M. Goresky; R. MacPherson Intersection homology. II, Invent. Math., Volume 72 (1983), pp. 77-129

[7] M. Lesch Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Teubner-Verlag, Germany, 1997

[8] U. Ludwig A comparison theorem between two complexes on singular spaces, J. Reine Angew. Math. (2016) (in press) | DOI

[9] M. Varghese; D. Quillen Superconnections, Thom classes, and equivariant differential forms, Topology, Volume 25 (1986), pp. 85-110

Cité par Sources :

Commentaires - Politique