Orthogonal bundles on curves and theta functions
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, p. 1405-1418

Let be the moduli space of principal SO r -bundles on a curve C, and the determinant bundle on . We define an isomorphism of H 0 (,) onto the dual of the space of r-th order theta functions on the Jacobian of C. This isomorphism identifies the rational map || * defined by the linear system || with the map |rΘ| which associates to a quadratic bundle (E,q) the theta divisor Θ E . The two components + and - of are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for Sp 2r -bundles.

Soient l’espace des modules des fibrés SO r -principaux sur une courbe C, et le fibré déterminant sur . Nous définissons un isomorphisme de H 0 (,) sur le dual de l’espace des fonctions thêta du r-ième ordre sur la Jacobienne de C. Cet isomorphisme identifie l’application rationnelle || * définie par le système linéaire || avec l’application |rΘ| qui associe à un fibré quadratique (E,q) le diviseur thêta Θ E . Les deux composantes + et - de sont envoyées sur les sous-espaces de fonctions paires et impaires respectivement. Finalement nous discutons le problème analogue pour les fibrés symplectiques.

DOI : https://doi.org/10.5802/aif.2216
Classification:  14H60
Keywords: Principal bundles, orthogonal bundles, symplectic bundles, theta divisors, generalized theta functions, Verlinde formula, strange duality
@article{AIF_2006__56_5_1405_0,
     author = {Beauville, Arnaud},
     title = {Orthogonal bundles on curves and theta functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {5},
     year = {2006},
     pages = {1405-1418},
     doi = {10.5802/aif.2216},
     mrnumber = {2273860},
     zbl = {1114.14021},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_5_1405_0}
}
Beauville, Arnaud. Orthogonal bundles on curves and theta functions. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1405-1418. doi : 10.5802/aif.2216. http://www.numdam.org/item/AIF_2006__56_5_1405_0/

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