Asymptotics for Bergman kernels for high powers of complex line bundles, based on joint works with B. Berndtsson and R. Berman

Sjöstrand, Johannes

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 22, 8 p.

Nous discutons l’asymptotique des noyaux de Bergman pour des puissances élevées de fibrés de droites, d’après deux travaux récents avec B.Berndtsson et R. Berman.

[1] B. Berndtsson, R. Berman, J. Sjöstrand, In preparation.

[2] R. Berman, J. Sjöstrand, In preparation.

[3] R. Berman, Bergman kernels and local holomorphic Morse inequalities, Math. Z. 248(2)(2004), 325–344. | MR | Zbl

[4] J.M. Bismut, Demailly’s asymptotic Morse inequalities, a heat equation proof, J. Funct. Anal., 72(1987), 263–278. | Zbl

[5] P. Bleher, B. Shiffman, S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142(2)(2000), 351–395. | MR | Zbl

[6] T. Bouche, Convergence de la métrique de Fubini Study d’un fibré linéaire positif, Ann. Inst. Fourier, 40(1)(1990), 117-130. | Numdam | Zbl

[7] L. Boutet de Monvel, V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, 99. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. | MR | Zbl

[8] L. Boutet de Monvel, J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Astérisque 34-35 (1976), 123–164. | Numdam | MR | Zbl

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

[10] L. Charles, Berezin-Toeplitz operators, a semi-classical approach, CMP 239(2003),1–28. | MR | Zbl

[11] X. Dai, K. Liu, X. Ma, On the asymptotic expansion of Bergman kernel, Preprint and CRAS 339(2004), 193–198. | MR | Zbl

[12] M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc Lecture Notes Series 268, Cambridge Univ. Press 1999. | MR | Zbl

[13] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26(1974), 1–65. | MR | Zbl

[14] L. Hörmander, An introduction to complex analysis in several variables, van Nostrand, (1966), 1967. | MR | Zbl

[15] Z. Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Am. J. Math., 122(2)(2000), 235–273. | MR | Zbl

[16] Z. Lu, G. Tian, The log term of Szegö kernel, Duke Math. J. 125(2)(2004), 351-387. | MR | Zbl

[17] X. Ma, Marinescu, The spin${}^c$ Dirac operator on high tensor powers of a line bundle, Math. Z. 240(3)(2002), 651–664. | MR | Zbl

[18] A. Melin, J. Sjöstrand, Fourier integral operators with complex valued phase functions, Springer LNM, 459. | MR | Zbl

[19] A. Melin, J. Sjöstrand, Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem, CPDE, 1(4)(1976), 313–400. | MR | Zbl

[20] A. Melin, J. Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods and Appl. of Anal. 9(2)(2002), 177–238. | MR | Zbl

[21] A. Menikoff, J. Sjöstrand, On the eigenvalues of a class of hypoelliptic operators, Math. Ann. 235(1978), 55–85. | MR | Zbl

[22] A. Menikoff, J. Sjöstrand, The eigenvalues of hypoelliptic operators III, the non-semibounded case, J. d’Analyse Math. 35(1979), 123–150. | Zbl

[23] W. Ruan, Canonical coordintes and Bergman metrics, Comm. Anal. Geom., 6(1998), 589–631. | MR | Zbl

[24] G. Tian, On a set of polarized Kähler metrics, J. Diff. Geom., 32(1990), 99–130. | MR | Zbl

[25] R.O. Wells, Differential analysis on complex manifolds, Graduate texts in mathematics 65, Springer 1980 | MR | Zbl

[26] S. Zelditch, Szegö kernels and a theorem of Tian, IMRN 1998(6), 317–331. | MR | Zbl