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Nonabelian theta functions of positive genus


Author: Arzu Boysal
Journal: Proc. Amer. Math. Soc. 136 (2008), 4201-4209
MSC (2000): Primary 14H60; Secondary 22E65
DOI: https://doi.org/10.1090/S0002-9939-08-09467-7
Published electronically: July 24, 2008
MathSciNet review: 2431033
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Abstract: Let $ \mathcal{C}_g$ be a smooth projective irreducible curve over $ \mathbb{C}$ of genus $ g \geq 1$ and let $ \{p_1,\dots, p_s\}$ be a set of distinct points on $ \mathcal{C}_g$. We fix a nonnegative integer $ \ell$ and denote by $ M_g(\underline{p},\underline{\lambda})$ the moduli space of parabolic semistable vector bundles of rank $ r$ on $ \mathcal{C}_g$ with trivial determinant and fixed parabolic structure of type $ \underline{\lambda}=(\lambda_1,\dots, \lambda_s)$ at $ \underline{p}=(p_1,\dots, p_s)$, where each weight $ \lambda_i$ is in $ P_{\ell}(\mathrm{SL}(r))$. On $ M_g(\underline{p},\underline{\lambda})$ there is a canonical line bundle $ \mathcal{L}(\underline{\lambda}, \ell)$, whose global sections are called generalized parabolic $ \mathrm{SL}(r)$-theta functions of order $ \ell$. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.


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Additional Information

Arzu Boysal
Affiliation: Université Paris 6, case 7012, 2 place Jussieu, 75251 Paris cedex 05, France
Email: boysal@math.jussieu.fr, arzu.boysal@boun.edu.tr

DOI: https://doi.org/10.1090/S0002-9939-08-09467-7
Keywords: Theta functions, parabolic vector bundles, factorization rules, fusion product, PRV, saturation conjecture
Received by editor(s): August 22, 2007
Received by editor(s) in revised form: November 16, 2007
Published electronically: July 24, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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