×

Equivariant Lefschetz number of differential operators. (English) Zbl 1205.32020

Authors’ abstract: Let \(G\) be a compact Lie group acting on a compact complex manifold \(M\) by holomorphic transformations. We prove a trace density formula for the \(G\)-Lefschetz number of a holomorphic differential operator on \(M\). We generalize the recent results of M. Engeli and the first author [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 623–655 (2008; Zbl 1163.32009)] to orbifolds.

MSC:

32L99 Holomorphic fiber spaces
32W50 Other partial differential equations of complex analysis in several variables
58J35 Heat and other parabolic equation methods for PDEs on manifolds
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
19L10 Riemann-Roch theorems, Chern characters
53C55 Global differential geometry of Hermitian and Kählerian manifolds

Citations:

Zbl 1163.32009
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alev J., Farinati M., Lambre T., Solotar A.: Homologie des invariants d’une algèbre de Weyl sous l’action d’une groupe fini. J. Algebra 232, 564–577 (2000) · Zbl 1002.16005 · doi:10.1006/jabr.2000.8406
[2] Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators, vol. 298. Springer, Grundlehren (1992) · Zbl 0744.58001
[3] Brylinski J., Getzler E.: The homology of algebras of pseudodifferential symbols and the noncommutative residue. K-Theory 1(4), 385–403 (1987) · Zbl 0646.58026 · doi:10.1007/BF00539624
[4] Cartan, H.: Sur les groupes de transformation analytiques. Actualiés Scientifiques et Industrielles, vol. 198, Paris (1935) · JFM 61.0370.02
[5] Cartan H.: OEuvres, vol. I. In: Remmert, R., Serre, J.-P. (eds) Works, vol. I, pp. 474–523. Springer, Berlin (1979)
[6] Duistermaat, J.J.: The heat kernel Lefschetz fixed point formula for the spin-c Dirac operator. In: Progress in Nonlinear Differential Equations and their Applications, vol. 18. Birkhäuser, Boston (1996) · Zbl 0858.58045
[7] Fedosov B.: On G-trace and G-index in deformation quantization, Conference Moshé Flato 1999 (Dijon). Lett. Math. Phys. 52(1), 29–49 (2002) · Zbl 0998.53058 · doi:10.1023/A:1007601802484
[8] Engeli M., Felder G.: A Riemann–Roch–Hirzebruch formula for traces of differential operators. Ann. Scient. Éc. Norm. Sup, 4 e série, t 41, 621–653 (2008)
[9] Feigin B., Felder G., Shoikhet B.: Hochschild cohomology of the Weyl algebra and traces in deformation quantization. Duke Math. J. 127(3), 487–517 (2005) · Zbl 1106.53055 · doi:10.1215/S0012-7094-04-12733-2
[10] Langer, A.: Logarithmic orbifold Euler numbers of surfaces with applications. In: Proceedings of the London Mathematical Society, vol. 3(86), 358–396 (2003) · Zbl 1052.14037
[11] Musson I.: Differential operators on toric varieties. J. Pure Appl. Algebra 95(E), 303–315 (1994) · Zbl 0824.14044 · doi:10.1016/0022-4049(94)90064-7
[12] Neumaier N., Pflaum M., Posthuma H., Tang X.: Homology of formal deformations of proper étale Lie groupoids. J. Reine Angew. Math 593, 117–168 (2006) · Zbl 1246.53120 · doi:10.1515/CRELLE.2006.031
[13] Pflaum M.J., Posthuma H., Tang X.: An algebraic index theorem for orbifolds. Adv. Math. 210, 83–121 (2007) · Zbl 1119.58014 · doi:10.1016/j.aim.2006.05.018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.