Bergman kernels and symplectic reduction
Astérisque, no. 318 (2008) , 162 p.
@book{AST_2008__318__R1_0,
author = {Ma, Xiaonan and Zhang, Weiping},
title = {Bergman kernels and symplectic reduction},
series = {Ast\'erisque},
publisher = {Soci\'et\'e math\'ematique de France},
number = {318},
year = {2008},
zbl = {1171.32001},
mrnumber = {2473876},
language = {en},
url = {http://www.numdam.org/item/AST_2008__318__R1_0/}
}
TY  - BOOK
AU  - Ma, Xiaonan
AU  - Zhang, Weiping
TI  - Bergman kernels and symplectic reduction
T3  - Astérisque
PY  - 2008
DA  - 2008///
IS  - 318
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2008__318__R1_0/
UR  - https://zbmath.org/?q=an%3A1171.32001
UR  - https://www.ams.org/mathscinet-getitem?mr=2473876
LA  - en
ID  - AST_2008__318__R1_0
ER  -
%0 Book
%A Ma, Xiaonan
%A Zhang, Weiping
%T Bergman kernels and symplectic reduction
%S Astérisque
%D 2008
%N 318
%I Société mathématique de France
%G en
%F AST_2008__318__R1_0
Ma, Xiaonan; Zhang, Weiping. Bergman kernels and symplectic reduction. Astérisque, no. 318 (2008), 162 p. http://numdam.org/item/AST_2008__318__R1_0/

[1] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundl. Math. Wiss. Band 298, Springer-Verlag, Berlin, 1992. | Zbl | MR

[2] J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators : two heat equation proofs, Invent. Math. 83 (1986), no. 1, 91-151. | DOI | EuDML | Zbl | MR

[3] J.-M. Bismut, Koszul complexes, harmonic oscillators, and the Todd class, J. Amer. Math. Soc. 3 (1990), no. 1, 159-256, With an appendix by the author and C. Soulé. | DOI | Zbl | MR

[4] J.-M. Bismut, Equivariant immersions and Quillen metrics, J. Differential Geom. 41 (1995), no. 1, 53-157. | DOI | Zbl | MR

[5] J.-M. Bismut, Holomorphic families of immersions and higher analytic torsion forms, Astérisque (1997), no. 244, viii+275. | Numdam | Zbl | MR

[6] J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms, Comm. Math. Phys. 115 (1988), no. 1, 79-126. | DOI | Zbl | MR

[7] J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics, Publ. Math. Inst. Hautes Études Sci. (1991), no. 74, ii+298 pp. (1992). | Numdam | EuDML | Zbl | MR

[8] J.-M. Bismut and É. Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), no. 2, 355-367. | DOI | Zbl | MR

[9] M. Bordemann and E. Meinrenken and M. Schlichenmaier, Toeplitz quantization of Kähler manifolds and $gl\left(N\right),N\to \infty$ limits, Comm. Math. Phys. 165 (1994), no. 2, pp. 281-296. | DOI | Zbl | MR

[10] D. Borthwick and A. Uribe, Almost complex structures and geometric quantization, Math. Res. Lett. 3 (1996), no. 6, 845-861. | DOI | Zbl | MR

D. Borthwick and A. Uribe, Almost complex structures and geometric quantization, Erratum : 5 (1998), 211-212. | Zbl | MR

[11] L. Boutet De Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ, 1981. | Zbl | MR

[12] L. Boutet De Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Journées : Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, Astérisque, No. 34-35, pp. 123-164. . | EuDML | Zbl | MR

[13] M. Braverman, Vanishing theorems on covering manifolds, Contemp. Math., vol. 231, Amer. Math. Soc, Providence, RI, 1999, pp. 1-23. | DOI | Zbl | MR

[14] D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, MA, 1999, pp. 1-23. | Zbl | MR

[15] L. Charles, Toeplitz operators and Hamiltonian torus action, J. Funct. Anal. 236 (2006), 299-350. | DOI | Zbl | MR

[16] J. Chazarain and A. Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981. | Zbl | MR

[17] X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), 1-41; announced in C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193-198. | DOI | Zbl | MR

[18] S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479-522. | DOI | Zbl | MR

[19] H. Fegan, Introduction to compact Lie groups, Series in Pure Mathematics, vol. 13, World Scientific Publishing Co. Inc., River Edge, NJ, 1991. | DOI | MR | Zbl

[20] V. Guillemin, Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys. 35 (1995), no. 1, 85-89. | DOI | Zbl | MR

[21] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515-538. | DOI | EuDML | Zbl | MR

[22] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. | Zbl | MR

[23] A. V. Karabegov and M. Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49-76. | Zbl | MR

[24] H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. | Zbl | MR

[25] Z. Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235-273. | DOI | Zbl | MR

[26] X. Ma, Orbifolds and analytic torsions, Trans. Amer. Math. Soc. 357 (2005), 2205-2233. | DOI | Zbl | MR

[27] X. Ma and G. Marinescu, The ${\mathrm{Spin}}^{c}$ Dirac operator on high tensor powers of a line bundle, Math. Z. 240 (2002), no. 3, 651-664. | DOI | Zbl | MR

[28] X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 493-498. | DOI | Zbl | MR

X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, The full version: math.DG/0411559, Adv. Math. 217 (2008), 1756-1815. | DOI | Zbl | MR

[29] X. Ma and G. Marinescu, The first coefficients of the asymptotic expansion of the Bergman kernel of the spin${}^{c}$ Dirac operator, Internat. J. Math. 17 (2006), 737-759. | DOI | Zbl | MR

[30] X. Ma and G. Marinescu, Toeplitz operators on symplectic manifolds, Preprint (2005), J. Geom. Anal., 18 (2008), 565-611. | DOI | Zbl | MR

[31] X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics 254, Birkhäuser Boston, Inc.., Boston, MA 2007, 422 pp. | Zbl | MR

[32] X. Ma and W. Zhang, Bergman kernels and symplectic reduction, C. R. Math. Acad. Sci. Paris 341 (2005), 297-302. | DOI | Zbl | MR

[33] X. Ma and W. Zhang, Toeplitz quantization and symplectic reduction, Differential Geometry and Physics. Eds. M.-L. Ge and Weiping Zhang, Nankai Tracts in Mathematics Vol. 10, World Scientific, (2006), 343-349. | DOI | Zbl | MR

[34] E. Meinrenken, On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc. 9 (1996), no. 2, 373-389. | DOI | Zbl | MR

[35] E. Meinrenken, Symplectic surgery and the ${\mathrm{Spin}}^{c}$-Dirac operator, Adv. Math. 134 (1998), no. 2, 240-277. | DOI | Zbl | MR

[36] R. Paoletti, Moment maps and equivariant Szegö kernels, J. Symplectic Geom. 2 (2003), 133-175. | DOI | Zbl | MR

[37] R. Paoletti, The Szegö kernel of a symplectic quotient, Adv. Math. 197 (2005), 523-553. | DOI | Zbl | MR

[38] W. Ruan, Canonical coordinates and Bergmann metrics, Comm. Anal. Geom. 6 (1998), 589-631. | DOI | Zbl | MR

[39] M. Schlichenmaier, Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 289-306. | DOI | Zbl | MR

[40] B. Shiffman and S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181-222. | Zbl | MR

[41] M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. | Zbl | MR

[42] M. E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. | Zbl | MR

M. E. Taylor, Partial differential equations. I, II, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. | Zbl | MR

[43] G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99-130. | DOI | Zbl | MR

[44] Y. Tian and W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), no. 2, 229-259. | DOI | Zbl | MR

[45] M. Vergne, Multiplicities formula for geometric quantization. I, II, Duke Math. J. 82 (1996), no. 1, 143-179, 181-194. | DOI | Zbl | MR

[46] S.-T. Yau, Nonlinear analysis in geometry. Enseign. Math. (2) 33 (1987), no. 1-2, 109-158. | Zbl | MR

[47] S. Zelditch, Szegö kernels and a theorem of Tian, Intern. Math. Res. Notices (1998), no. 6, 317-331. | DOI | Zbl | MR

[48] W. Zhang, Holomorphic quantization formula in singular reduction, Commun. Contemp. Math. 1 (1999), no. 3, 281-293. | DOI | Zbl | MR