The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle
Bismut, Jean-Michel ; Vasserot, E.
Annales de l'Institut Fourier, Tome 40 (1990), p. 835-848 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {40},
     year = {1990},
     pages = {835-848},
     doi = {10.5802/aif.1237},
     mrnumber = {92b:58237},
     zbl = {0711.32015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 835-848. doi : 10.5802/aif.1237. http://gdmltest.u-ga.fr/item/AIF_1990__40_4_835_0/

[B1] J. M. Bismut, The index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math., 83 (1987), 91-151. | MR 87g:58117 | Zbl 0592.58047

[B2] J. M. Bismut, Demailly's asymptotic Morse inequalities: a heat equation proof, J. Funct. Anal., 72 (1987), 263-278. | MR 886814 | MR 88j:58131 | Zbl 0649.58030

[BGS1] J. M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles. II, Comm. Math. Phys., 115 (1988), 79-126. | MR 89g:58192b | Zbl 0651.32017

[BGS2] J. M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles. III, Comm. Math. Phys., 115 (1988), 301-351. | MR 931666 | MR 89g:58192c | Zbl 0651.32017

[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 1016875 | MR 91c:58141 | Zbl 0687.32023

[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 961475 | MR 89k:32058 | Zbl 0651.32019

[Ge] E. Getzler, Inégalités asymptotiques de Demailly pour les fibrés vectoriels, C.R. Acad. Sci., Série I. Math., 304 (1987), 475-478. | MR 894572 | MR 88j:32040 | Zbl 0614.32022

[GrH] P. Griffiths, J. Harris, Principles of algebraic geometry, New York, Wiley, 1978. | MR 507725 | MR 80b:14001 | Zbl 0408.14001

[K] S. Kobayashi, Differential geometry of complex vector bundles, Iwanami Shoten and Princeton University Press, 1987. | MR 909698 | MR 89e:53100 | Zbl 0708.53002

[LP] J. Le Potier, Théorèmes d'annulation en cohomologie, C.R. Acad. Sci. Paris, Série A, 276 (1976), 535-537. | MR 49 #7482 | Zbl 0249.32021

[Q] D. Quillen, Superconnections and the Chern character, Topology, 24 (1985), 89-95. | MR 86m:58010 | Zbl 0569.58030

[RS] D. B. Ray, I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math., 98 (1973), 154-177. | MR 52 #4344 | Zbl 0267.32014

[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 38 #6220 | Zbl 0159.15504