On the asymptotic expansion of Bergman kernel

Xianzhe Dai; Kefeng Liu; Xiaonan Ma

doi:10.1016/j.crma.2004.05.011

Differential Geometry

On the asymptotic expansion of Bergman kernel
[Sur le développement asymptotique du noyau de Bergman.]

Présenté par : Jean-Michel Bismut

Xianzhe Dai ¹ ; Kefeng Liu ^2,³ ; Xiaonan Ma ⁴

¹ Department of Mathematics, UCSB, California, CA 93106, USA
² Center of Mathematical Science, Zhejiang University, China
³ Department of Mathematics, UCLA, California, CA 90095-1555, USA
⁴ Centre de mathématiques, CNRS UMR 7640, École polytechnique, 91128 Palaiseau cedex, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 193-198.

Résumé
Abstract

On étudions les développements asymptotiques du noyau de la chaleur et de Bergman de l'opérateur de Dirac ${spin}^{c}$ associé à une puissance grande d'un fibré en droites positif.

We study the asymptotics of the Bergman kernel and the heat kernel of the ${spin}^{c}$ Dirac operator on high tensor powers of a line bundle.

Reçu le : 2004-05-14
Accepté le : 2004-05-18
Publié le : 2004-07-23

DOI : 10.1016/j.crma.2004.05.011

Affiliations des auteurs :

Xianzhe Dai ¹ ; Kefeng Liu ^{2, 3} ; Xiaonan Ma ⁴

¹ Department of Mathematics, UCSB, California, CA 93106, USA
² Center of Mathematical Science, Zhejiang University, China
³ Department of Mathematics, UCLA, California, CA 90095-1555, USA
⁴ Centre de mathématiques, CNRS UMR 7640, École polytechnique, 91128 Palaiseau cedex, France

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     title = {On the asymptotic expansion of {Bergman} kernel},
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     pages = {193--198},
     publisher = {Elsevier},
     volume = {339},
     number = {3},
     year = {2004},
     doi = {10.1016/j.crma.2004.05.011},
     language = {en},
}

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Xianzhe Dai; Kefeng Liu; Xiaonan Ma. On the asymptotic expansion of Bergman kernel. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 193-198. doi : 10.1016/j.crma.2004.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.011/

Bibliographie
Cité par

[1] N. Berline; E. Getzler; M. Vergne Heat Kernels and Dirac Operators, Springer-Verlag, 1992

[2] J.-M. Bismut; G. Lebeau Complex immersions and Quillen metrics, Publ. Math. IHES, Volume 74 (1991), pp. 1-297

[3] J.-M. Bismut; E. Vasserot The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle, Commun. Math. Phys., Volume 125 (1989), pp. 355-367

[4] L. Boutet de Monvel; J. Sjöstrand Sur la singularité des noyaux de Bergman et de Szegö, Astérisque, Volume 34–35 (1976), pp. 123-164

[5] 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

[6] X. Dai; K. Liu; X. Ma On the asymptotic expansion of Bergman kernel | arXiv

[7] S.K. Donaldson Scalar curvature and projective embeddings. I, J. Differential Geom., Volume 59 (2001) no. 3, pp. 479-522

[8] T. Kawasaki The Riemann–Roch theorem for V-manifolds, Osaka J. Math., Volume 16 (1979), pp. 151-159

[9] Z. Lu On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math., Volume 122 (2000) no. 2, pp. 235-273

[10] X. Ma, Orbifolds and analytic torsions, Preprint

[11] X. Ma; G. Marinescu The ${spin}^{c}$ Dirac operator on high tensor powers of a line bundle, Math. Z., Volume 240 (2002) no. 3, pp. 651-664

[12] W. Ruan Canonical coordinates and Bergman metrics, Comm. Anal. Geom., Volume 6 (1998), pp. 589-631

[13] G. Tian On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., Volume 32 (1990), pp. 99-130

[14] X. Wang, Thesis, 2002

[15] S. Zelditch Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices, Volume 6 (1998), pp. 317-331

Hendrik Herrmann; Chin-Yu Hsiao; Xiaoshan Li Szegö kernel asymptotic expansion on strongly pseudoconvex CR manifolds with S1 action, International Journal of Mathematics, Volume 29 (2018) no. 09, p. 1850061 | DOI:10.1142/s0129167x18500611
Zhiwei Wang Asymptotics of orbifold Bergman kernels: mixed curvature case, Archiv der Mathematik, Volume 101 (2013) no. 5, p. 485 | DOI:10.1007/s00013-013-0575-3
Michael R. Douglas; Semyon Klevtsov Bergman Kernel from Path Integral, Communications in Mathematical Physics, Volume 293 (2010) no. 1, p. 205 | DOI:10.1007/s00220-009-0915-0
William D. Kirwin Coherent states in geometric quantization, Journal of Geometry and Physics, Volume 57 (2007) no. 2, p. 531 | DOI:10.1016/j.geomphys.2006.04.007
XIAONAN MA; GEORGE MARINESCU THE FIRST COEFFICIENTS OF THE ASYMPTOTIC EXPANSION OF THE BERGMAN KERNEL OF THESpincDIRAC OPERATOR, International Journal of Mathematics, Volume 17 (2006) no. 06, p. 737 | DOI:10.1142/s0129167x06003667

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