# A note on Berezin-Toeplitz quantization of the Laplace operator

Complex Manifolds (2015)

- Volume: 2, Issue: 1, page 131-139, electronic only
- ISSN: 2300-7443

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topAlberto Della Vedova. "A note on Berezin-Toeplitz quantization of the Laplace operator." Complex Manifolds 2.1 (2015): 131-139, electronic only. <http://eudml.org/doc/275930>.

@article{AlbertoDellaVedova2015,

abstract = {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.},

author = {Alberto Della Vedova},

journal = {Complex Manifolds},

keywords = {Hodge manifold; Berezin-Toeplitz quantization map},

language = {eng},

number = {1},

pages = {131-139, electronic only},

title = {A note on Berezin-Toeplitz quantization of the Laplace operator},

url = {http://eudml.org/doc/275930},

volume = {2},

year = {2015},

}

TY - JOUR

AU - Alberto Della Vedova

TI - A note on Berezin-Toeplitz quantization of the Laplace operator

JO - Complex Manifolds

PY - 2015

VL - 2

IS - 1

SP - 131

EP - 139, electronic only

AB - 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.

LA - eng

KW - Hodge manifold; Berezin-Toeplitz quantization map

UR - http://eudml.org/doc/275930

ER -

## References

top- [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. Zbl0813.58026
- [3] M. Engliˇs. Weighted Bergman kernels and quantization. Comm. Math. Phys. 227 (2002), no. 2, 211–241. Zbl1010.32002
- [4] J. Fine. Quantisation and the Hessian of the Mabuchi energy. Duke Math. J. 161, 14 (2012), 2753–2798. [WoS] Zbl1262.32024
- [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. Zbl1254.32036
- [6] A. V. Karabegov and M. Schlichenmaier. Identification of Berezin-Toeplitz deformation quantization. J. Reine Angew. Math. 540 (2001), 49–76. Zbl0997.53067
- [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). Zbl1135.32001
- [9] X. Ma and G. Marinescu. Berezin-Toeplitz quantization on K¨ahler manifolds. J. Reine Angew. Math. 662 (2012), 1–56. Zbl1251.47030
- [10] J. H.Rawnsley. Coherent states and K¨ahler manifolds. Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 403–415. Zbl0387.58002
- [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. Zbl0719.53044
- [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. Zbl0706.53036
- [15] S. Zelditch. Szeg˝o kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331. [Crossref] Zbl0922.58082

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