Comptes Rendus
no. 5
Differential Geometry
Bergman kernels and symplectic reduction
[Noyaux de Bergman et réduction symplectique]

Présenté par : Jean-Michel Bismut

Xiaonan Ma1 ; Weiping Zhang2
1 Centre de mathématiques, UMR 7640 du CNRS, École polytechnique, 91128 Palaiseau cedex, France
2 Nankai Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China
Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 297-302.

Nous annonçons des résultats sur le développement asymptotique du noyau de Bergman G-invariant de l'opérateur de Dirac spinc associé à une puissance tendant vers l'infini d'un fibré en droites positif sur une variété symplectique compacte.

We present several results concerning the asymptotic expansion of the invariant Bergman kernel of the spinc Dirac operator associated with high tensor powers of a positive line bundle on a compact symplectic manifold.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.07.009
Xiaonan Ma 1 ; Weiping Zhang 2

1 Centre de mathématiques, UMR 7640 du CNRS, École polytechnique, 91128 Palaiseau cedex, France
2 Nankai Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China 
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Xiaonan Ma; Weiping Zhang. Bergman kernels and symplectic reduction. Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 297-302. doi : 10.1016/j.crma.2005.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.009/

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[2] X. Dai; K. Liu; X. Ma On the asymptotic expansion of Bergman kernel, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 3, pp. 193-198 (The full version: J. Differential Geom., preprint) | arXiv

[3] V. Guillemin; S. Sternberg Geometric quantization and multiplicities of group representations, Invent. Math., Volume 67 (1982) no. 3, pp. 515-538

[4] X. Ma; G. Marinescu The Spinc Dirac operator on high tensor powers of a line bundle, Math. Z., Volume 240 (2002) no. 3, pp. 651-664

[5] X. Ma; G. Marinescu Generalized Bergman kernels on symplectic manifolds, C.R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 7, pp. 493-498 (The full version) | arXiv

[6] X. Ma, W. Zhang, Bergman kernels and symplectic reduction, preprint

[7] R. Paoletti Moment maps and equivariant Szegö kernels, J. Symplectic Geom., Volume 2 (2003), pp. 133-175

[8] R. Paoletti The Szegö kernel of a symplectic quotient, Adv. Math. (2005) | arXiv

[9] Y. Tian; W. Zhang An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg, Invent. Math., Volume 132 (1998) no. 2, pp. 229-259

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