Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-bv7lh Total loading time: 0.143 Render date: 2021-06-16T07:29:44.313Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Torelli theorem for the moduli space of framed bundles

Published online by Cambridge University Press:  26 November 2009

I. BISWAS
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. e-mail: indranil@math.tifr.res.in
T. GÓMEZ
Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Serrano 113bis, 28006 Madrid, Spain and Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: tomas.gomez@mat.csic.es, vicente.munoz@mat.ucm.es
V. MUÑOZ
Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Serrano 113bis, 28006 Madrid, Spain and Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: tomas.gomez@mat.csic.es, vicente.munoz@mat.ucm.es

Abstract

Let X be an irreducible smooth complex projective curve of genus g ≥ 2, and let xX be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Exr is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Atiyah, M. F.Vector bundles over an elliptic curve. Proc. London Math. Soc. (3) 7 (1957), 414452.CrossRefGoogle Scholar
[2]Atiyah, M. F. and Bott, R.The Yang-Mills equations over Riemann surfaces. Roy. Soc. London Philos. Trans. Ser. A 308 (1982), 523615.CrossRefGoogle Scholar
[3]Biswas, I., Brambila-Paz, L. and Newstead, P. E.Stability of projective Poincaré and Picard bundles. Bull. Lond. Math. Soc. 41 (2009), no. 3, 458472.CrossRefGoogle Scholar
[4]Biswas, I. and Gómez, T.Simplicity of stable principal sheaves. Bull. Lond. Math. Soc. 40 (2008), 163171.CrossRefGoogle Scholar
[5]Bradlow, S. B., García-Prada, O., Mercat, V., Muñoz, V. and Newstead, P. E.On the geometry of moduli spaces of coherent systems on algebraic curves. Internat. J. Math. 18 (2007), 411453.CrossRefGoogle Scholar
[6]Brambila-Paz, L., Grzegorczyk, I. and Newstead, P. E.Geography of Brill-Noether loci for small slopes. J. Alg. Geom. 6 (1997), 645669.Google Scholar
[7]Drézet, J.-M. and Narasimhan, M. S.Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97 (1989), 5394.CrossRefGoogle Scholar
[8]Hitchin, N.Stable bundles and integrable systems. Duke Math. J. 54 (1987), 91114.CrossRefGoogle Scholar
[9]Huybrechts, D. and Lehn, M.Framed modules and their moduli. Internat. J. Math. 6 (1995), 297324.CrossRefGoogle Scholar
[10]Huybrechts, D. and Lehn, M. The geometry of moduli spaces of sheaves. Aspects of Mathematics E31, (Vieweg, Braunschweig/Wiesbaden 1997).Google Scholar
[11]Kouvidakis, A. and Pantev, T.The automorphism group of the moduli space of semistable vector bundles. Math. Ann. 302 (1995), 225268.CrossRefGoogle Scholar
[12]Luna, D. Slices étales. Sur les groupes algébriques. pp. 81–105. Bull. Soc. Math. France, Paris, Memoire 33 Soc. Math. France, Paris, 1973.Google Scholar
[13]Maruyama, M.Openness of a family of torsion free sheaves. Jour. Math. Kyoto Univ. 16 (1976), 627637.CrossRefGoogle Scholar
[14]Narasimhan, M. S. and Ramanan, S. Geometry of Hecke cycles I, C. P. Ramanujam – a tribute. pp. 291345. Tata Inst. Fund. Res. Studies in Math., 8 (Springer, 1978).Google Scholar
[15]Seshadri, C. S.Fibrés vectoriels sur les courbes algébriques. Astérisque 96 (1982), 209 pp.Google Scholar
[16]Simpson, C.Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. IHES 79 (1994), 47129.CrossRefGoogle Scholar
[17]Tyurin, A. N.Classification of vector bundles over an algebraic curve of arbitrary genus. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 657688.Google Scholar
2
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Torelli theorem for the moduli space of framed bundles
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Torelli theorem for the moduli space of framed bundles
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Torelli theorem for the moduli space of framed bundles
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *