A Torelli theorem for moduli spaces of principal bundles over a curve
type: Item
instance of: scholarly article
title: A Torelli theorem for moduli spaces of principal bundles over a curve
zbMATH Open document ID: 1268.14010
author: Indranil Biswas
author: Norbert Hoffmann
publication date: 2012-08-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.4061
reviewed by: Jesse Leo Kass
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14C34
zbMATH DE Number: 6064513
zbMATH Keywords: principal bundle
zbMATH Keywords: moduli space
zbMATH Keywords: Torelli theorem
MaRDI profile type: Publication
OpenAlex ID: W2963088237
arXiv ID: 1003.4061
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DOI: 10.5802/AIF.2700
published in: Annales de l’institut Fourier
review text: The authors prove a Torelli-type theorem for the moduli space of principal \(G\)-bundles. The classical Torelli Theorem states that two curves must be isomorphic if their principally polarized Jacobians, or polarized moduli spaces of topologically trivial line bundles, are isomorphic. The authors of the present paper generalize this result by replacing the Jacobian with a more general moduli space.NEWLINENEWLINEThey replace the Jacobian by the moduli space \(\text{M}_{G}^{d}(X)\) of polystable principal \(G\)-bundle of fixed topological type \(d\) on a smooth complex projective curve \(X\). Suppose that \(G\) and \(G'\) are two non-abelian connected reductive groups. The main theorem of this paper states that two curves \(X, X'\) of genus at least \(3\) are isomorphic if the modules spaces \(\text{M}_{G}^{d}(X)\) and \(\text{M}_{G'}^{d'}(X')\) are isomorphic.NEWLINENEWLINEThe authors prove the theorem by showing that \(X\) can be reconstructed from \(\text{M}^{d}_{G}(X)\). Most of the paper is devoted to proving this when \(\text{M}^{d}_{G}(X)\) contains a strictly semistable point. When no such point exists, \(G\) must be of a special form: The quotient of \(G\) by its center must be isomorphic to a product \(\text{PGL}_{n_1} \times \dots \times \text{PGL}_{n_r}\) of projective linear groups. In this special case, the authors reconstruct a suitable moduli space of semistable vector bundles and then use a previous Torelli-type theorem of \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math. (2) 101, 391--417 (1975; Zbl 0314.14004)] and \textit{A. N. Tyurin} [Math. USSR, Izv. 4, (1970), 343--370 (1970; Zbl 0225.14011)].NEWLINENEWLINEWhen \(\text{M}^{d}_{G}(X)\) contains a strictly semistable point, the authors reconstruct the Jacobian with its principal polarization and then use the classical Torelli Theorem. To reconstruct the Jacobian, the authors prove several results on the singular locus and the strictly semistable locus of \(\text{M}_{G}^{d}(X)\). For e…
description: scientific article; zbMATH DE number 6064513