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Quantization of the Laplacian operator on vector bundles, I

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Abstract

Let (E, h) be a holomorphic, Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.

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Notes

  1. Alternatively, one can prove by induction that \(\mathrm {tr}(A^2)=\frac{ 4 (k+i+1)! (i+1)! (k-i) ! i! }{ (k!)^2 (2i+2)!}\).

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Acknowledgments

The first author is very grateful to Joel Fine for illuminating conversations on the subject of balanced embeddings throughout the years. Furthermore the second author wants to thank him for the numerous conversations they had and for generously sharing his ideas with him. The work of the first author has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). The first author was also partially supported by supported by the ANR Project EMARKS, Decision No ANR-14-CE25-0010. The second author was supported by an AFR Ph.D. Grant from the Fonds National de la Recherche Luxembourg, and acknowledges travel support from the Communauté française de Belgique via an ARC and from the Belgian federal government via the PAI “Dygest”.

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Authors and Affiliations

  1. Institut de Mathématiques de Marseille I2M, UMR 7373, Aix - Marseille Université, 39 rue F. Joliot-Curie, 13453, Marseille Cedex 13, France

    Julien Keller

  2. Département de Mathématique, Université libre de Bruxelles, CP 218, Boulevard du Triomphe, 1050, Brussels, Belgium

    Julien Meyer

  3. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Reza Seyyedali

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  1. Julien Keller
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  2. Julien Meyer
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  3. Reza Seyyedali
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Correspondence to Julien Keller.

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Keller, J., Meyer, J. & Seyyedali, R. Quantization of the Laplacian operator on vector bundles, I. Math. Ann. 366, 865–907 (2016). https://doi.org/10.1007/s00208-015-1355-0

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