Orthogonal bundles on curves and theta functions

Beauville, Arnaud

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1405-1418.

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

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

[1] Alexeev, A.; Meinrenken, E.; Woodward, C. Formulas of Verlinde type for non-simply connected groups (Preprint, math.SG/0005047) | MR

[2] Beauville, A. Vector bundles on curves and theta functions Preprint, math.AG/0502179, Proc. of the conf. "Moduli spaces and arithmetic geometry" (Kyoto, 2004). Advanced studies in pure math, to appear

[3] Beauville, A. Fibrés de rang $2$ sur les courbes, fibré déterminant et fonctions thêta II, Bull. Soc. Math. France, Volume 119 (1991) no. 3, pp. 259-291 | Numdam | MR | Zbl

[4] Beauville, A. Conformal blocks, Fusion rings and the Verlinde formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) (Israel Math. Conf. Proc.), Volume 9 (1996), pp. 75-96 | MR | Zbl

[5] Beauville, A.; Laszlo, Y.; Sorger, C. The Picard group of the moduli of $G$-bundles on a curve, Compositio Math., Volume 112 (1998) no. 2, pp. 183-216 | DOI | MR | Zbl

[6] Beauville, A.; Narasimhan, M.S.; Ramanan, S. Spectral curves and the generalised theta divisor, J. Reine Angew. Math., Volume 398 (1989), pp. 169-179 | DOI | MR | Zbl

[7] Bourbaki, N. Groupes et algèbres de Lie. Chap. VI, Hermann, Paris, 1968 | MR | Zbl

[8] Drezet, J.-M.; Narasimhan, M.S. Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math., Volume 97 (1989) no. 1, pp. 53-94 | DOI | MR | Zbl

[9] Dynkin, E. Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Translations (II), Volume 6 (1957), pp. 111-244 | Zbl

[10] Kumar, S.; Narasimhan, M.S. Picard group of the moduli spaces of $G$-bundles, Math. Ann., Volume 308 (1997) no. 1, pp. 155-173 | DOI | MR | Zbl

[11] Laszlo, Y. À propos de l’espace des modules de fibrés de rang 2 sur une courbe, Math. Ann., Volume 299 (1994) no. 4, pp. 597-608 | DOI | MR | Zbl

[12] Laszlo, Y.; Sorger, C. The line bundles on the moduli of parabolic $G$-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4), Volume 30 (1997) no. 4, pp. 499-525 | Numdam | MR | Zbl

[13] Mumford, D. On the equations defining abelian varieties, I, Invent. Math., Volume 1 (1966), pp. 287-354 | DOI | MR | Zbl

[14] Mumford, D. Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100 | MR | Zbl

[15] Oxbury, W.; Wilson, S. Reciprocity laws in the Verlinde formulae for the classical groups, Trans. Amer. Math. Soc., Volume 348 (1996) no. 7, pp. 2689-2710 | DOI | MR | Zbl

[16] Ramanan, S. Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. Math. Sci., Volume 90 (1981) no. 2, pp. 151-166 | DOI | MR | Zbl

[17] Serre, J.-P. Revêtements à ramification impaire et thêta-caractéristiques, C. R. Acad. Sci. Paris Sér. I Math., Volume 311 (1990) no. 9, pp. 547-552 | MR | Zbl

[18] Sorger, C. On moduli of $G$-bundles of a curve for exceptional $G$, Ann. Sci. École Norm. Sup. (4), Volume 32 (1999) no. 1, pp. 127-133 | Numdam | MR | Zbl