Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space
Annales de l'Institut Fourier, Volume 47 (1997) no. 3, pp. 885-914.

The moduli space of stable vector bundles over a moving curve is constructed, and on this a generalized Weil-Petersson form is constructed. Using the local Riemann-Roch formula of Bismut-Gillet-Soulé it is shown that the generalized Weil-Petersson form is the curvature of the determinant line bundle, equipped with the Quillen metric, for a vector bundle on the fiber product of the universal moduli space with the universal curve.

Nous construisons l’espace de module des fibrés vectoriels stables sur une courbe mobile, et construisons sur cet espace une forme de Weil-Petersson généralisée. En utilisant la formule de Riemann-Roch locale de Bismut-Gillet-Soulé, nous montrons que la forme de Weil-Petersson généralisée est la courbure du fibré déterminant, muni de la métrique de Quillen, pour un fibré vectoriel sur le produit fibré de l’espace de module universel avec la courbe universelle.

@article{AIF_1997__47_3_885_0,
     author = {Biswas, Indranil},
     title = {Determinant bundle over the universal moduli space of vector bundles over the {Teichm\"uller} space},
     journal = {Annales de l'Institut Fourier},
     pages = {885--914},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {3},
     year = {1997},
     doi = {10.5802/aif.1584},
     mrnumber = {98i:32025},
     zbl = {0873.32017},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1584/}
}
TY  - JOUR
TI  - Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space
JO  - Annales de l'Institut Fourier
PY  - 1997
DA  - 1997///
SP  - 885
EP  - 914
VL  - 47
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1584/
UR  - https://www.ams.org/mathscinet-getitem?mr=98i:32025
UR  - https://zbmath.org/?q=an%3A0873.32017
UR  - https://doi.org/10.5802/aif.1584
DO  - 10.5802/aif.1584
LA  - en
ID  - AIF_1997__47_3_885_0
ER  - 
%0 Journal Article
%T Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space
%J Annales de l'Institut Fourier
%D 1997
%P 885-914
%V 47
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1584
%R 10.5802/aif.1584
%G en
%F AIF_1997__47_3_885_0
Biswas, Indranil. Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space. Annales de l'Institut Fourier, Volume 47 (1997) no. 3, pp. 885-914. doi : 10.5802/aif.1584. https://aif.centre-mersenne.org/articles/10.5802/aif.1584/

[AB] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Tran. Roy. Soc. Lond., A308 (1982), 523-615. | MR | Zbl

[BGS1] J.M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles I. Bott-Chern forms and analytic torsion, Commun. Math. Phy., 115 (1988), 49-78. | MR | Zbl

[BGS2] J.M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles II. Direct images and Bott-Chern forms, Commun. Math. Phy., 115 (1988), 79-126. | MR | Zbl

[BGS3] J.M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles III. Quillen metrices on holomorphic determinants, Commun. Math. Phy., 115 (1988) 301-351. | MR | Zbl

[BR] I. Biswas, N. Raghavendra, Determinants of parabolic bundles on Riemann surface, Proc. Indian Acad. Sci., 103 (1993), 41-71. | MR | Zbl

[FS] A. Fujiki, G. Schumacher, The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metric, Publ. R.I.M.S. Kyoto Univ., 26 (1990), 101-183. | MR | Zbl

[G] W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), 200-225. | MR | Zbl

[K] N.M. Katz, An overview of Deligne's work on Hilbert's twenty-first problem, Proc. Symp. Pure Math., 28 (1976), 537-557. | MR | Zbl

[KM] F. Knudsen, D. Mumford, The projectivity of the moduli space of stable curves I. Preliminaries on "det" and "div", Math. Scand., 39 (1976), 19-55. | MR | Zbl

[Ko] S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Math. Soc. of Japan, Iwanami Schoten Pub. and Princeton Univ. Press (1987). | MR | Zbl

[KT] F. Kamber, P. Tondeur, Foliated bundles and characteristic classes, Lec. Notes in Math. 493, Springer-Verlag, Berlin-Heidelberg-New York, (1975). | MR | Zbl

[MS] V. Mehta, C. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann., 248 (1980), 205-239. | MR | Zbl

[NS] M.S. Narasimhan, C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math., 82 (1965), 540-567. | MR | Zbl

[Q] D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. An. Appl., 19 (1985), 31-34. | MR | Zbl

[S1] G. Schumacher, Moduli of polarized Kähler manifolds, Math. Ann., 269 (1984), 137-144. | MR | Zbl

[ST] G. Schumacher, M. Toma, Moduli of Kähler manifolds equipped with Hermite-Einstein vector bundles, Rev. Roumaine Math. Pures Appl., 38 (1993), 703-719. | MR | Zbl

[ZT] P. Zograf, L. Takhtadzhyan, A local index theorem for families of ∂-operators on Riemann surfaces, Russian Math. Surveys, 42-6 (1987), 169-190. | MR | Zbl

Cited by Sources: