A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve
type: Item
instance of: scholarly article
title: A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve
zbMATH Open document ID: 1314.14061
DOI: 10.1142/S0129167X14500475
author: Insong Choe
author: G. H. Hitching
published in: International Journal of Mathematics
publication date: 2014-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.0834
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 6327999
zbMATH Keywords: vector bundle
zbMATH Keywords: algebraic curve
zbMATH Keywords: symplectic and orthogonal structures
MaRDI profile type: Publication
arXiv ID: 1204.0834
cites work: Lagrangian subbundles of symplectic bundles over a curve
cites work: Subbundles of symplectic and orthogonal vector bundles over curves
cites work: Moduli of rank 4 symplectic vector bundles over a curve of genus 2
cites work: Counting maximal subbundles via Gromov-Witten invariants
cites work: A generalisation of Nagata's theorem on ruled surfaces.
cites work: Zur Klassifikation von Regelmannigfaltigkeiten
cites work: Universal families of extensions
cites work: Maximal subbundles of rank two vector bundles on curves
cites work: Deformations of the moduli space of vector bundles over an algebraic curve
cites work: Stable principal bundles on a compact Riemann surface
cites work: Moduli spaces of orthogonal and symplectic bundles over an algebraic curve
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review text: Let \(X\) be a smooth projective algebraic curve over \({\mathbb C}\) of genus \(g\geq2\) and let \(V\) be a vector bundle on \(X\) of rank \(2\). The Segre invariant \(s(V)\) of \(V\) is defined by \(s(V):=\min\{\deg V-2\deg L\}\), the minimum being taken over all line subbundles \(L\) of \(V\). This invariant provides a stratification of the moduli spaces of semistable vector bundles on \(X\), which was investigated by \textit{H. Lange} and \textit{M. S. Narasimhan} [Math. Ann. 266, 55--72 (1983; Zbl 0507.14005)]. A generalisation to higher rank was described by \textit{L. Brambila-Paz} and \textit{H. Lange} [J. Reine Angew. Math. 494, 173--187 (1998; Zbl 0919.14016)] and by \textit{B. Russo} and \textit{M. Teixidor i Bigas} [J. Algebr. Geom. 8, No. 3, 483--496 (1999; Zbl 0942.14013)].NEWLINENEWLINEIn the present paper, the authors consider an analogue of the Segre invariant for symplectic and orthogonal bundles \(V\) of rank \(2n\). The invariant is defined as \(t(V):= -2\max\{\deg E: E\text{ a rank } n \text{ isotropic subbundle of }V\}\). To state the results, let \(\mathrm{SU}_X(2n,{\mathcal O}_X)\) denote the moduli space of semistable bundles on \(X\) with trivial determinant and write \({\mathcal M}S_X(2n)\) (respectively, \({\mathcal M}O_X(2n)\)) for the sublocus of \(\mathrm{SU}_X(2n,{\mathcal O}_X)\) of bundles admitting a symplectic (respectively, orthogonal) structure. (In fact \({\mathcal M}S_X(2n)\) is isomorphic to the moduli space of semistable symplectic bundles, but the corresponding result does not hold in the orthogonal case.) In the orthogonal case, when \(n\geq2\), there are two components, distinguished by the value of the second Stiefel-Whitney class, which are denoted by \({\mathcal M}O_X(2n)^\pm\). The invariant \(t\) induces stratifications of these spaces and we write \({\mathcal M}S_X(2n,t):=\{V\in{\mathcal M}S_X(2n,t): t(V)=t)\}\) with similar notations in the orthogonal case.NEWLINENEWLINEIn the symplectic case, the main result (Theo…
reviewed by: P. E. Newstead
description: scientific article; zbMATH DE number 6327999