Differential Geometry
Circle actions and Z/k-manifolds
[Actions du cercle et Z/k variétés]

Présenté par : Jean-Michel Bismut

Zhang, Weiping1
1 Nankai Institute of Mathematics, Nankai University, Tianjin 300071, PR China
Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 57-60.

On établit un théorème d'indice S1-équivariant pour les opérateurs de Dirac sur des Z/k variétés. On donne une application de ce résultat, qui généralise le théorème d'Atiyah–Hirzebruch sur les actions de S1 aux Z/k variétés.

We establish an S1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah–Hirzebruch vanishing theorem for S1-actions on closed spin manifolds to the case of Z/k-manifolds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00279-6
Zhang, Weiping 1

1 Nankai Institute of Mathematics, Nankai University, Tianjin 300071, PR China 
@article{CRMATH_2003__337_1_57_0,
     author = {Zhang, Weiping},
     title = {Circle actions and $ \mathrm{Z}\mathrm{/k}$-manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {57--60},
     publisher = {Elsevier},
     volume = {337},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00279-6},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00279-6/}
}
TY  - JOUR
AU  - Zhang, Weiping
TI  - Circle actions and $ \mathrm{Z}\mathrm{/k}$-manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 57
EP  - 60
VL  - 337
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00279-6/
DO  - 10.1016/S1631-073X(03)00279-6
LA  - en
ID  - CRMATH_2003__337_1_57_0
ER  - 
%0 Journal Article
%A Zhang, Weiping
%T Circle actions and $ \mathrm{Z}\mathrm{/k}$-manifolds
%J Comptes Rendus. Mathématique
%D 2003
%P 57-60
%V 337
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00279-6/
%R 10.1016/S1631-073X(03)00279-6
%G en
%F CRMATH_2003__337_1_57_0
Zhang, Weiping. Circle actions and $ \mathrm{Z}\mathrm{/k}$-manifolds. Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 57-60. doi : 10.1016/S1631-073X(03)00279-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00279-6/

[1] Atiyah, M.F.; Hirzebruch, F. Spin manifolds and groups actions (Haefliger, A.; Narasimhan, R., eds.), Essays on Topology and Related Topics, Mémoirée dédié à Georges de Rham, Springer-Verlag, 1970, pp. 18-28

[2] Atiyah, M.F.; Patodi, V.K.; Singer, I.M. Spectral asymmetry and Riemannian geometry I, Proc. Cambridge Philos. Soc., Volume 77 (1975), pp. 43-69

[3] Bismut, J.-M.; Lebeau, G. Complex immersions and Quillen metrics, Publ. Math. IHES, Volume 74 (1991), pp. 1-297

[4] Bismut, J.-M.; Zhang, W. Real embeddings and eta invariant, Math. Ann., Volume 295 (1993), pp. 661-684

[5] Dai, X.; Zhang, W. Real embeddings and the Atiyah–Patodi–Singer index theorem for Dirac operators, Asian J. Math., Volume 4 (2000), pp. 775-794

[6] Freed, D.S. Z/k-manifolds and families of Dirac operators, Invent. Math., Volume 92 (1988), pp. 243-254

[7] Freed, D.S.; Melrose, R.B. A mod k index theorem, Invent. Math., Volume 107 (1992), pp. 283-299

[8] Liu, K.; Ma, X.; Zhang, W. Rigidity and vanishing theorems in K-theory, Comm. Anal. Geom., Volume 11 (2003), pp. 121-180

[9] Tian, Y.; Zhang, W. Quantization formula for symplectic manifolds with boundary, Geom. Funct. Anal., Volume 9 (1999), pp. 596-640

[10] Witten, E. Fermion quantum numbers in Kaluza–Klein theory (Jackiw, R. et al., eds.), Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field theory and the Fundamental Problems of Physics, MIT Press, 1985, pp. 227-277

Cité par Sources :