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Orthogonal bundles on curves and theta functions. (English) Zbl 1114.14021

Let \(G\) be an almost simple complex Lie group. The paper under review is devoted to the study of certain principal \(G\)-bundles on a curve \(C\) of genus \(g \geq 2\). Let \(\mathcal{M}_G\) be the moduli space of such semi-stable vector bundles. The Picard group of each component \(\mathcal{M}^{\bullet}_G\) of \(\mathcal{M}_G\) is infinite cyclic and let \(\mathcal{L}^{\bullet}\) be the generator. Mainly, the author considers the cases \(G=\text{ SO}_r\) and \(\text{ Sp}_{2r}\) and the rational map \(\varphi_G:\mathcal{M}_G \dashrightarrow | \mathcal{L}^{\bullet}| ^*\) given by global sections of \(\mathcal{L}^{\bullet}\). Let \(J^{g-1}\) be the component of the Picard variety of \(C\) parametrizing line bundles of degree \(g-1\). The first part of the paper is devoted to exploring the relation between \(\varphi_{\text{ SO}_r}\) and degree \(r\) theta functions on \(J^{g-1}\). Let \(\mathcal{M}_{\text{ SO}_r}^{\pm}\) be the two components of \(\mathcal{M}_{\text{ SO}_r}\) and \(| r\Theta| ^{\pm}\) the subspaces of respectively even and odd theta functions. We recall the well-known theta-map \(\theta: {M}_{\text{ SO}_r} \dashrightarrow | r\Theta| \) that associates to an orthogonal bundle \((E,q)\) the divisor \[ \Theta_E:= \{L \in J^{g-1}| H^0(C,E\otimes L)\neq 0\}\in | r\Theta| . \] The main result of this part of the paper is the following: there are canonical isomorphisms \[ | \mathcal{L}^{\pm}_{\text{ SO}_r}| ^* \overset{\sim}{\rightarrow} | r\Theta| ^{\pm} \] which identify \(\varphi_{\text{ SO}_r}^{\pm}: \mathcal{M}_{\text{ SO}_r}^{\pm} \dashrightarrow | \mathcal{L}^{\pm}_{\text{ SO}_r}| ^*\) with the map \(\theta^{\pm}: \mathcal{M}_{\text{ SO}_r}^{\pm} \dashrightarrow | r\Theta| ^{\pm}\) induced by \(\theta\). This is equivalent to showing that the pull-back map \[ \theta^*: H^0(J^{g-1}, \mathcal{O}(r\Theta))^* \rightarrow H^0(\mathcal{M}_{\text{ SO}_r},\mathcal{L}_{\text{ SO}_r}) \] is an isomorphism. The injectivity is proven by restricting to a subvariety of \(\mathcal{M}_{\text{ SO}_r}\) and the surjectivity via the Verlinde formula. In the second section the author considers the same question for the symplectic group \(\text{ Sp}_{2r}\). The main difference is that in this case the theta-map does not involve \(J^{g-1}\) but the moduli space \(\mathcal{N}\) of semistable rank 2 vector bundles on \(C\) with canonical determinant. Let \(\mathcal{L_N}\) be the determinant bundle on \(\mathcal{N}\); then for a general \((E,\sigma)\in \mathcal{M}_{\text{ Sp}_{2r}}\) the reduced subvariety \[ \Delta_E=\{F\in \mathcal{N}| H^0(E\otimes F)\neq 0\} \] is a divisor on \(\mathcal{N}\) that belongs to the linear system \(| \mathcal{L_N}^r| \). This defines a map \(\mathcal{M}_{\text{ Sp}_{2r}} \dashrightarrow | \mathcal{L_N}^r| \). The author conjectures (and can show in some cases) that this map should coincide, up to a canonical isomorphism, with \(\varphi_{\text{ Sp}_{2r}}\).

MSC:

14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
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References:

[1] Alexeev, A.; Meinrenken, E.; Woodward, C., Formulas of Verlinde type for non-simply connected groups
[2] Beauville, A., Vector bundles on curves and theta functions · Zbl 1115.14025
[3] Beauville, A., Fibrés de rang \(2\) sur les courbes, fibré déterminant et fonctions thêta II, Bull. Soc. Math. France, 119, 3, 259-291 (1991) · Zbl 0756.14017
[4] Beauville, A., Conformal blocks, Fusion rings and the Verlinde formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 9, 75-96 (1996) · Zbl 0848.17024
[5] Beauville, A.; Laszlo, Y.; Sorger, C., The Picard group of the moduli of \(G\)-bundles on a curve, Compositio Math., 112, 2, 183-216 (1998) · Zbl 0976.14024
[6] Beauville, A.; Narasimhan, M. S.; Ramanan, S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math., 398, 169-179 (1989) · Zbl 0666.14015
[7] Bourbaki, N., Groupes et algèbres de Lie. Chap. VI (1968) · Zbl 0483.22001
[8] Drezet, J.-M.; Narasimhan, M. S., Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math., 97, 1, 53-94 (1989) · Zbl 0689.14012
[9] Dynkin, E., Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Translations (II), 6, 111-244 (1957) · Zbl 0077.03404
[10] Kumar, S.; Narasimhan, M. S., Picard group of the moduli spaces of \(G\)-bundles, Math. Ann., 308, 1, 155-173 (1997) · Zbl 0884.14004
[11] Laszlo, Y., À propos de l’espace des modules de fibrés de rang 2 sur une courbe, Math. Ann., 299, 4, 597-608 (1994) · Zbl 0846.14011
[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), 30, 4, 499-525 (1997) · Zbl 0918.14004
[13] Mumford, D., On the equations defining abelian varieties, I, Invent. Math., 1, 287-354 (1966) · Zbl 0219.14024
[14] Mumford, D., Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), 29-100 (1970) · Zbl 0198.25801
[15] Oxbury, W.; Wilson, S., Reciprocity laws in the Verlinde formulae for the classical groups, Trans. Amer. Math. Soc., 348, 7, 2689-2710 (1996) · Zbl 0902.14031
[16] Ramanan, S., Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. Math. Sci., 90, 2, 151-166 (1981) · Zbl 0512.14018
[17] Serre, J.-P., Revêtements à ramification impaire et thêta-caractéristiques, C. R. Acad. Sci. Paris Sér. I Math., 311, 9, 547-552 (1990) · Zbl 0742.14030
[18] Sorger, C., On moduli of \(G\)-bundles of a curve for exceptional \(G\), Ann. Sci. École Norm. Sup. (4), 32, 1, 127-133 (1999) · Zbl 0969.14016
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