Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions

doi:10.1017/CBO9781107416000.018

17 - Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions

Published online by Cambridge University Press: 05 January 2015

R. Paoletti

Edited by

Christopher D. Hacon ,

Mircea Mustaţă and

Mihnea Popa

Show author details

R. Paoletti: Affiliation:
Università degli Studi di Milano Bicocca
Christopher D. Hacon: Affiliation:
University of Utah
Mircea Mustaţă: Affiliation:
University of Michigan, Ann Arbor
Mihnea Popa: Affiliation:
University of Illinois, Chicago

Book contents

Get access

Summary

Abstract

We consider a natural variant of Berezin–Toeplitz quantization of compact Kähler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szegö and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient. When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin–Toeplitz quantization. This continues previous work on neardiagonal scaling asymptotics of equivariant Szegö kernels in the presence of Hamiltonian torus actions.

Dedicated to Rob Lazarsfeld on the occasion of his 60th birthday

1 Introduction

The object of this paper are the asymptotics of Szegö and Toeplitz operators in a non-standard version of the Berezin–Toeplitz quantization of a complex projective Kähler manifold (M, J, ω).

In Berezin–Toeplitz quantization, one typically adopts as “quantum spaces” the Hermitian spaces H0 (M, A⊗k of global holomorphic sections of higher powers of the polarizing line bundle (A, h); here (A, h) is a positive, hence ample, Hermitian holomorphic line bundle on M. Quantum observables, on the contrary, correspond to Toeplitz operators associated with real C∞ functions on M.

Here we assume given a Hamiltonian action μM of the circle group U(1) = T1 on M, with positive moment map Φ, and admitting a metric-preserving linearization to A. It is then natural to replace the spaces H0 (M, A⊗k with certain new “quantum spaces” which arise by decomposing the Hardy space associated with A into isotypes for the action; these are generally not spaces of sections of powers of A. One is thus led to consider analogues of the usual constructs of Berezin–Toeplitz quantization. In particular, it is interesting to investigate how the symplectic geometry of the underlying action, encapsulated in Φ, influences the semiclassical asymptotics in this quantizationscheme.

Type: Chapter
Information: Recent Advances in Algebraic Geometry
A Volume in Honor of Rob Lazarsfeld’s 60th Birthday
, pp. 321 - 369

DOI: https://doi.org/10.1017/CBO9781107416000.018 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

[1] Ali, S.T. and Engliš, M.Quantization methods: A guide for physicists and analysts. Rev. Math. Phys. 17(4) (2005) 391–490.Google Scholar

[2] Berezin, F. A.General concept of quantization. Comm. Math. Phys. 40 (1975) 153–174.Google Scholar

[3] Bleher, P., Shiffman, B., and Zelditch, S.Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142 (2000) 351–395.Google Scholar

[4] Bordemann, M., Meinrenken, E., and Schlichenmaier, M.Toeplitz quantization of Kähler manifolds and gl(N), N → ∞ limits. Comm. Math. Phys. 165(2) (1994) 281–296.Google Scholar

[5] Boutet de Monvel, L. and Guillemin, V.The Spectral Theory of Toeplitz Operators. Annals of Mathematics Studies, Vol. 99. Princeton, NJ: Princeton University Press, 1981.

[6] Boutet de Monvel, L. and Sjöstrand, J.Sur la singularité des noyaux de Bergman etde Szegö. Astérisque 34/35 (1976) 123–164.Google Scholar

[7] Cahen, M., Gutt, S., and Rawnsley, J.Quantization of Kähler manifolds. I. Geometric interpretation of Berezin's quantization. J. Geom. Phys. 7 (1990) 45–62.Google Scholar

[8] Cahen, M., Gutt, S., and Rawnsley, J.Quantization of Kahlermanifolds. II. Trans. Amer. Math. Soc. 337 (1993) 73–98.Google Scholar

[9] Calabi, E.Isometric imbedding of complex manifolds. Ann. Math. (2) 58 (1953) 123.Google Scholar

[10] Catlin, D.The Bergman kernel and a theorem of Tian. In Analysis and Geometry in Several Complex Variables (Katata, 1997), pp. 1–23. Trends in Mathematics. Boston: Birkhauser, 1999.

[11] Charles, L.Berezin–Toeplitz operators, a semi-classical approach. Comm. Math. Phys. 239(1&2) (2003) 128.Google Scholar

[12] Engliš, M.Berezin quantization and reproducing kernels on complex domains. Trans. Amer. Math. Soc. 348(2) (1996) 411–479.Google Scholar

[13] Engliš, M.The asymptotics of a Laplace integral on a Kähler manifold. J. Reine Angew. Math. 528 (2000) 139.Google Scholar

[14] Guillemin, V.Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35 (1995) 85–89.Google Scholar

[15] Guillemin, V., Ginzburg, V., and Karshon, Y.Moment Maps, Cobordisms, and Hamiltonian Group Actions. Appendix J by Maxim Braverman. Mathematical Surveys and Monographs, Vol. 98. Providence, RI: American Mathematical Society, 2002.

[16] Guillemin, V. and Sternberg, S.Geometric quantization and multiplicities of group representations. Invent. Math. 67(3) (1982) 515–538.Google Scholar

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

[18] Krantz, S. and Parks, H.A Primer of Real Analytic Functions, 2nd edn. Boston: Birkhauser, 2002.

[19] Loi, A.The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics. Int. J. Geom. Methods Mod. Phys. 1(3) (2004) 253–263.Google Scholar

[20] Loi, A.A Laplace integral on a Kahler manifold and Calabi's diastasis function. Diff. Geom. Appl. 23 (2005) 55–66.Google Scholar

[21] Loi, A.A Laplace integral, the T–Y–Z expansion, and Berezin's transform on a Kähler manifold. Int. J. Geom. Methods Mod. Phys. 2(3) (2005) 359–371.Google Scholar

[22] Lu, Z.On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch. Amer. J. Math. 122(2) (2000) 235–273.Google Scholar

[23] Ma, X. and Marinescu, G.Holomorphic Morse Inequalities and Bergman Kernels. Progress and in Mathematics, Vol. 254. Basel: Birkhäuser, 2007.

[24] Ma, X. and Marinescu, G.Toeplitz operators on symplectic manifolds. J. Geom. Anal. 18(2) (2008)565–611.Google Scholar

[25] Ma, X. and Marinescu, G.Berezin–Toeplitz quantization and its kernel expansion. In Geometry and Quantization, pp. 125–166. Travaux Mathématiques, Vol. 19. University of Luxembourg, Luxembourg, 2011.

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

[27] Melin, A. and Sjöstrand, J.Fourier integral operators with complex-valued phase functions. In Fourier Integral Operators and Partial Differential Equations International Colloquium, University of Nice, Nice, 1974), pp. 120–223. Lecture Notes in Mathematics, Vol. 459. Berlin: Springer-Verlag, 1975.

[28] Paoletti, R.Asymptotics of Szegö kernels under Hamiltonian torus actions. Israel J. Math. 191 (2012) 363–403.Google Scholar

[29] Schlichenmaier, M.Berezin–Toeplitz quantization for compact Kähler manifolds. An introduction. In Geometry and Quantization, pp. 97–124. Travaux Mathématiques, Vol. 19. University of Luxenbourg, Luxembourg, 2011.

[30] Shiffman, B. and Zelditch, S.Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544 (2002) 181–222.Google Scholar

[31] Tian, G.On a set of polarized Kähler metrics on algebraic manifolds. J. Diff. Geom. 32 (1990) 99–130.Google Scholar

[32] Tian, G.Canonical metrics in Kähler geometry. Notes taken by Meike Akveld. Lectures in Mathematics ETH Zürich. Basel: Birkhauser, 2000.

[33] Xu, H.An explicit formula for the Berezin star product. Lett. Math. Phys. 101(3) (2012) 239–264.Google Scholar

[34] Zelditch, S.Index and dynamics of quantized contact transformations. Ann. Inst. Fourier (Grenoble) 47 (1997) 305–363.Google Scholar

[35] Zelditch, S.Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6 (1998) 317–331.Google Scholar