Orthogonal bundles on curves and theta functions

Arnaud Beauville[1]

  • [1] Université de Nice Laboratoire J.A. Dieudonné Parc Valrose 06108 Nice Cedex 2 (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1405-1418
  • ISSN: 0373-0956

Abstract

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Let ℳ be the moduli space of principal 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 ⇢ | ℒ | * defined by the linear system | ℒ | with the map ℳ ⇢ | r Θ | which associates to a quadratic bundle ( E , q ) the theta divisor Θ 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 Sp 2 r -bundles.

How to cite

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Beauville, Arnaud. "Orthogonal bundles on curves and theta functions." Annales de l’institut Fourier 56.5 (2006): 1405-1418. <http://eudml.org/doc/10180>.

@article{Beauville2006,
abstract = {Let $\mathcal\{M\}$ be the moduli space of principal $\mathrm\{SO\}_r$-bundles on a curve $C$, and $\mathcal\{L\}$ the determinant bundle on $\mathcal\{M\}$. We define an isomorphism of $H^0(\mathcal\{ M\},\mathcal\{L\})$ onto the dual of the space of $r$-th order theta functions on the Jacobian of $C$. This isomorphism identifies the rational map $\mathcal\{M\}\dasharrow|\mathcal\{L\}|^*$ defined by the linear system $|\mathcal\{L\}|$ with the map $\mathcal\{M\}\dasharrow|r \Theta |$ which associates to a quadratic bundle $(E,q)$ the theta divisor $\Theta _E$. The two components $\mathcal\{M\}^+$ and $\mathcal\{M\}^-$ of $\mathcal\{ M\}$ are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for $\mathrm\{Sp\}_\{2r\}$-bundles.},
affiliation = {Université de Nice Laboratoire J.A. Dieudonné Parc Valrose 06108 Nice Cedex 2 (France)},
author = {Beauville, Arnaud},
journal = {Annales de l’institut Fourier},
keywords = {Principal bundles; orthogonal bundles; symplectic bundles; theta divisors; generalized theta functions; Verlinde formula; strange duality; vector bundles on curves; determinant bundle; moduli spaces},
language = {eng},
number = {5},
pages = {1405-1418},
publisher = {Association des Annales de l’institut Fourier},
title = {Orthogonal bundles on curves and theta functions},
url = {http://eudml.org/doc/10180},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Beauville, Arnaud
TI - Orthogonal bundles on curves and theta functions
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1405
EP - 1418
AB - Let $\mathcal{M}$ be the moduli space of principal $\mathrm{SO}_r$-bundles on a curve $C$, and $\mathcal{L}$ the determinant bundle on $\mathcal{M}$. We define an isomorphism of $H^0(\mathcal{ M},\mathcal{L})$ onto the dual of the space of $r$-th order theta functions on the Jacobian of $C$. This isomorphism identifies the rational map $\mathcal{M}\dasharrow|\mathcal{L}|^*$ defined by the linear system $|\mathcal{L}|$ with the map $\mathcal{M}\dasharrow|r \Theta |$ which associates to a quadratic bundle $(E,q)$ the theta divisor $\Theta _E$. The two components $\mathcal{M}^+$ and $\mathcal{M}^-$ of $\mathcal{ M}$ are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for $\mathrm{Sp}_{2r}$-bundles.
LA - eng
KW - Principal bundles; orthogonal bundles; symplectic bundles; theta divisors; generalized theta functions; Verlinde formula; strange duality; vector bundles on curves; determinant bundle; moduli spaces
UR - http://eudml.org/doc/10180
ER -

References

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