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Complex Manifolds

Ed. by Fino, Anna Maria

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A note on Berezin-Toeplitz quantization of the Laplace operator

Alberto Della Vedova
  • Corresponding author
  • Universit`a degli Studi di Milano-Bicocca. Dipartimento di Matematica e Applicazioni. Via Cozzi, 53 - 20125 Milano, Italy
  • Other articles by this author:
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Published Online: 2015-09-28 | DOI: https://doi.org/10.1515/coma-2015-0010


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.


  • [1] S. Donaldson. Scalar Curvature and Projective Embeddings, I. J. Diff. Geometry 59, (2001), 479–522. Google Scholar

  • [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. Google Scholar

  • [3] M. Engliˇs. Weighted Bergman kernels and quantization. Comm. Math. Phys. 227 (2002), no. 2, 211–241. Google Scholar

  • [4] J. Fine. Quantisation and the Hessian of the Mabuchi energy. Duke Math. J. 161, 14 (2012), 2753–2798. Web of ScienceGoogle Scholar

  • [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. Google Scholar

  • [6] A. V. Karabegov and M. Schlichenmaier. Identification of Berezin-Toeplitz deformation quantization. J. Reine Angew. Math. 540 (2001), 49–76. Google Scholar

  • [7] J. Keller, J. Meyer and R. Seyyedali. Quantization of the Laplacian operator on vector bundles I arXiv:1505.03836 [math.DG] Google Scholar

  • [8] X. Ma and G. Marinescu. Holomorphic Morse inequalities and Bergman kernels. Birkh¨auser (2007). Google Scholar

  • [9] X. Ma and G. Marinescu. Berezin-Toeplitz quantization on K¨ahler manifolds. J. Reine Angew. Math. 662 (2012), 1–56. Google Scholar

  • [10] J. H.Rawnsley. Coherent states and K¨ahler manifolds. Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 403–415. Google Scholar

  • [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. Google Scholar

  • [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]. Google Scholar

  • [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]. Google Scholar

  • [14] G. Tian. On a set of polarized K¨ahler metrics on algebraic manifolds J. Differential Geometry 32 (1990) 99–130. Google Scholar

  • [15] S. Zelditch. Szeg˝o kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331. CrossrefGoogle Scholar

About the article

Received: 2015-05-15

Accepted: 2015-09-23

Published Online: 2015-09-28

Citation Information: Complex Manifolds, Volume 2, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2015-0010.

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© 2015 Alberto Della Vedova. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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