Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.
@article{bwmeta1.element.doi-10_1515_coma-2015-0010,
author = {Alberto Della Vedova},
title = {A note on Berezin-Toeplitz quantization of the Laplace operator},
journal = {Complex Manifolds},
volume = {2},
year = {2015},
zbl = {1323.32014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0010}
}
Alberto Della Vedova. A note on Berezin-Toeplitz quantization of the Laplace operator. Complex Manifolds, Tome 2 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0010/
[1] S. Donaldson. Scalar Curvature and Projective Embeddings, I. J. Diff. Geometry 59, (2001), 479–522.
[2] M. Bordemann, E. Meinrenken and M. Schlichenmaier. Toeplitz quantization of K¨ahler manifolds and gl(N), N → 1 limits. Comm. Math. Phys. 165 (1994), no. 2, 281–296. | Zbl 0813.58026
[3] M. Engliˇs. Weighted Bergman kernels and quantization. Comm. Math. Phys. 227 (2002), no. 2, 211–241. | Zbl 1010.32002
[4] J. Fine. Quantisation and the Hessian of the Mabuchi energy. Duke Math. J. 161, 14 (2012), 2753–2798. [WoS] | Zbl 1262.32024
[5] A. Ghigi. On the approximation of functions on a Hodge manifold. Annales de la facult´e des sciences de Toulouse Math´ematiques 21, 4 (2012), 769–781. | Zbl 1254.32036
[6] A. V. Karabegov and M. Schlichenmaier. Identification of Berezin-Toeplitz deformation quantization. J. Reine Angew. Math. 540 (2001), 49–76. | Zbl 0997.53067
[7] J. Keller, J. Meyer and R. Seyyedali. Quantization of the Laplacian operator on vector bundles I arXiv:1505.03836 [math.DG]
[8] X. Ma and G. Marinescu. Holomorphic Morse inequalities and Bergman kernels. Birkh¨auser (2007). | Zbl 1135.32001
[9] X. Ma and G. Marinescu. Berezin-Toeplitz quantization on K¨ahler manifolds. J. Reine Angew. Math. 662 (2012), 1–56. | Zbl 1251.47030
[10] J. H.Rawnsley. Coherent states and K¨ahler manifolds. Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 403–415. | Zbl 0387.58002
[11] J. Rawnsley, M. Cahen and S. Gutt. Quantization of K¨ahler manifolds. I. Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7 (1990), no. 1, 45–62. | Zbl 0719.53044
[12] M. Schlichenmaier. Berezin-Toeplitz quantization and Berezin symbols for arbitrary compact K¨ahler manifolds. In ‘Proceedings of the XVIIth workshop on geometric methods in physics, Bia lowie˙za, Poland, July 3 – 10, 1998’ (M. Schlichenmaier, et. al. Eds.), Warsaw University Press, 45–56. arXiv:math/9902066v2 [math.QA].
[13] M. Schlichenmaier. Berezin-Toeplitz quantization and star products for compact K¨ahler manifolds. In ‘Mathematical Aspects of Quantization’, S. Evens, M. Gekhtman, B. C. Hall, X. Liu, C. Polini Eds. Contemporary Mathematics 583. AMS (2012). arXiv:1202.5927v3 [math.QA].
[14] G. Tian. On a set of polarized K¨ahler metrics on algebraic manifolds J. Differential Geometry 32 (1990) 99–130. | Zbl 0706.53036
[15] S. Zelditch. Szeg˝o kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331. [Crossref] | Zbl 0922.58082