Abstract
By the quantization condition compact quantizable Kähler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, singularities, and quotients appearing in geometric invariant theory are recalled.
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References
[] St. Berceanu and M. Schlichenmaier, Coherent state embeddings, polar divisors and Cauchy formulas, J. Geom. Phys. 34(2000), 337–360, math. QA/9902066.
F.A. Berezin, Quantization, Math. USSR-Izv. 8 (1974), 1109–1165.
M. Bordemann, J. Hoppe, P. Schaller, and M. Schlichenmaier, g/(ox) and geometric quantization, Commun. Math. Phys. 138 (1991), 209–244.
[] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of Kähler manifolds and gl(n), n -; oo limits, Commun. Math. Phys. 165 (1994), 281–296.
M. Bordemann and St. Waldmann, A Fedosov star product of the Wick type for Kähler manifolds, Lett. Math. Phys. 41 (1997), 243–253.
D. Eisenbud and J. Harris, The Geometry of Schemes, Springer, Berlin, 2000.
Ph. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley, New York, 1978.
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), 109–326.
[] J. Huebschmann, Singularities and Poisson geometry of certain representation spaces, this volume.
[] A. Karabegov and M. Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization, Mannheimer Manuskripte 253, math. QA/0006063, 2000.
A. Karabegov, Deformation quantization with separation of variables on a Kähler manifold, Commun. Math. Phys. 180 (1996), 745–755.
J. Lipman, Introduction to the resolution of singularities,in Algebraic Geometry (Arcata,1974), Proc. Symp. Pure Math. 29 (1975), 187–230.
D. Mumford, Geometric Invariant Theory, Springer, Berlin, 1982.
P.E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Springer, Berlin, 1978.
M. Schlichenmaier, Introduction to Riemann Surfaces,Algebraic Curves and Moduli Spaces, Lecture Notes in Physics 322, Springer, Berlin, 1990.
[] M. Schlichenmaier, Zwei Anwendungen algebraisch-geometrischer Methoden in der theoretischen Physik: Berezin-Toeplitz-Quantisierung und globale Algebren der zweidimensionalen konformen Feldtheorie, Habilitation thesis, 1996.
[] M. Schlichenmaier, Berezin-Toeplitz quantization of compact Kähler manifolds,in Quantization, Coherent States and Poisson Structures, Proc. XIV’th Workshop on Geometric Methods in Physics (Bialowieza, 1995), A. Strasburger, S.T. Ali, J.-P. Antoine, J.-P. Gazeau, and A. Odzijewicz, eds., 101–115, Polish Scientific Publishers (PWN), 1998, q-alg/9601016.
[] M. Schlichenmaier, Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, Conf. Mosh¨¦ Flato 1999 (Dijon, 1999), Vol. 2, G. Dito and D. Sternheimer, eds., 289–306, Kluwer, Dordrecht, 2000, math.QA/9910137.
[] M. Schlichenmaier, Berezin-Toeplitz quantization and Berezin transform, in Proc. Workshop on Asymptotic Properites of Time Evolutions in Classical and Quantum Systems (Bologna, 1999), math.QA/0009219, 2000.
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Schlichenmaier, M. (2001). Singular projective varieties and quantization. In: Landsman, N.P., Pflaum, M., Schlichenmaier, M. (eds) Quantization of Singular Symplectic Quotients. Progress in Mathematics, vol 198. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8364-1_10
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DOI: https://doi.org/10.1007/978-3-0348-8364-1_10
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