The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle
Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 835-848.

L’objet de cet article est de calculer le comportement asymptotique de la torsion analytique de Ray-Singer associée à la p-ième puissance symétrique d’un fibré vectoriel holomorphe Hermitien positif quand p tend vers +. Nous étendons ainsi notre résultat antérieur relatif aux fibrés en droites positifs.

The purpose of this paper is to calculate the asymptotics of the Ray-Singer analytic torsion associated with the p-th symmetric power of a holomorphic Hermitian positive vector bundle when p tends to +. We thus extend our previous results on positive line bundles.

@article{AIF_1990__40_4_835_0,
     author = {Bismut, Jean-Michel and Vasserot, E.},
     title = {The asymptotics of the {Ray-Singer} analytic torsion of the symmetric powers of a positive vector bundle},
     journal = {Annales de l'Institut Fourier},
     pages = {835--848},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {4},
     year = {1990},
     doi = {10.5802/aif.1237},
     mrnumber = {92b:58237},
     zbl = {0711.32015},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1237/}
}
TY  - JOUR
AU  - Bismut, Jean-Michel
AU  - Vasserot, E.
TI  - The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle
JO  - Annales de l'Institut Fourier
PY  - 1990
SP  - 835
EP  - 848
VL  - 40
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1237/
DO  - 10.5802/aif.1237
LA  - en
ID  - AIF_1990__40_4_835_0
ER  - 
%0 Journal Article
%A Bismut, Jean-Michel
%A Vasserot, E.
%T The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle
%J Annales de l'Institut Fourier
%D 1990
%P 835-848
%V 40
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1237/
%R 10.5802/aif.1237
%G en
%F AIF_1990__40_4_835_0
Bismut, Jean-Michel; Vasserot, E. The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle. Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 835-848. doi : 10.5802/aif.1237. http://www.numdam.org/articles/10.5802/aif.1237/

[B1] J. M. Bismut, The index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math., 83 (1987), 91-151. | EuDML | MR | Zbl

[B2] J. M. Bismut, Demailly's asymptotic Morse inequalities: a heat equation proof, J. Funct. Anal., 72 (1987), 263-278. | MR | Zbl

[BGS1] J. M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles. II, Comm. Math. Phys., 115 (1988), 79-126. | MR | Zbl

[BGS2] J. M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles. III, Comm. Math. Phys., 115 (1988), 301-351. | MR | Zbl

[BV] J. M. Bismut, E. Vasserot. The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys., 125 (1989), 355-367. | MR | Zbl

[De] J. P. Demailly, Vanishing theorems for tensor powers of a positive vector bundle. In Geometry and Analysis, T. Sunada, ed., pp. 86-106, Lecture Notes in Math. Berlin-Heidelberg-New York, Springer-Verlag, 1988. | MR | Zbl

[Ge] E. Getzler, Inégalités asymptotiques de Demailly pour les fibrés vectoriels, C.R. Acad. Sci., Série I. Math., 304 (1987), 475-478. | MR | Zbl

[GrH] P. Griffiths, J. Harris, Principles of algebraic geometry, New York, Wiley, 1978. | MR | Zbl

[K] S. Kobayashi, Differential geometry of complex vector bundles, Iwanami Shoten and Princeton University Press, 1987. | MR | Zbl

[LP] J. Le Potier, Théorèmes d'annulation en cohomologie, C.R. Acad. Sci. Paris, Série A, 276 (1976), 535-537. | MR | Zbl

[Q] D. Quillen, Superconnections and the Chern character, Topology, 24 (1985), 89-95. | MR | Zbl

[RS] D. B. Ray, I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math., 98 (1973), 154-177. | MR | Zbl

[Se] R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure and Appl. Math., Vol. 10, 288-307, Providence, Am. Math. Soc., (1967). | MR | Zbl

Cité par Sources :