Mathai, Melrose, et Singer ont introduit la notion d'opérateur elliptique projectif sur des variétés équipées d'un fibré d'Azumaya. Dans cette note, nous calculons les indices équivariants des opérateurs transversalement elliptiques qui s'obtiennent comme les tirés en arrière d'opérateurs elliptiques projectifs sur la variété qui trivialise le fibré d'Azumaya. Ces calculs généralisent la formule cohomologique de l'indice fractionnaire obtenue par Mathai–Melrose–Singer.
Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note, we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encompasses the fractional index formula of projective elliptic operator by Mathai–Melrose–Singer.
Accepté le :
Publié le :
@article{CRMATH_2016__354_12_1230_0, author = {Paradan, Paul-\'Emile}, title = {Index of projective elliptic operators}, journal = {Comptes Rendus. Math\'ematique}, pages = {1230--1235}, publisher = {Elsevier}, volume = {354}, number = {12}, year = {2016}, doi = {10.1016/j.crma.2016.10.018}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2016.10.018/} }
TY - JOUR AU - Paradan, Paul-Émile TI - Index of projective elliptic operators JO - Comptes Rendus. Mathématique PY - 2016 SP - 1230 EP - 1235 VL - 354 IS - 12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.10.018/ DO - 10.1016/j.crma.2016.10.018 LA - en ID - CRMATH_2016__354_12_1230_0 ER -
Paradan, Paul-Émile. Index of projective elliptic operators. Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1230-1235. doi : 10.1016/j.crma.2016.10.018. http://www.numdam.org/articles/10.1016/j.crma.2016.10.018/
[1] Elliptic Operators and Compact Groups, Lecture Notes in Mathematics, vol. 401, Springer-Verlag, Berlin, 1974
[2] The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math., Volume 124 (1996), pp. 11-49
[3] L'indice équivariant des opérateurs transversalement elliptiques, Invent. Math., Volume 124 (1996), pp. 51-101
[4] Heat Kernels and Dirac Operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004
[5] The index of elliptic operators over V-manifolds, Nagoya Math. J., Volume 9 (1981), pp. 135-157
[6] The index of projective families of elliptic operators, Geom. Topol., Volume 9 (2005), pp. 341-373
[7] Fractional analytic index, J. Differ. Geom., Volume 74 (2006), pp. 265-292
[8] Equivariant and fractional index of projective elliptic operators, J. Differ. Geom., Volume 78 (2008), pp. 465-473
[9] The index of projective families of elliptic operators: the decomposable case, Astérisque, Volume 328 (2009), pp. 255-296
[10] Index of transversally elliptic operators, Astérisque, Volume 328 (2009), pp. 297-338
[11] Index character associated with the projective Dirac operator, Proc. Amer. Math. Soc., Volume 141 (2013), pp. 2923-2932
Cité par Sources :