Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T05:07:02.106Z Has data issue: false hasContentIssue false

Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces

Published online by Cambridge University Press:  20 November 2018

Christian Gantz
Affiliation:
Mathematical Institute Oxford OX1 3LB UK
Brian Steer
Affiliation:
Mathematical Institute Oxford OX1 3LB UK
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Atiyah, M. F. and Bott, R., The Yang-Mills equation over Riemann surfaces. Philos. Trans. Roy. Soc. London A 308(1982), 5230615.Google Scholar
[2] Barth, W., Peters, C. and Van de Ven, A., Compact complex surfaces. Springer, Berlin-Heidelberg-New York, 1984.Google Scholar
[3] Bauer, S., Parabolic bundles, elliptic surfaces and SU(2)-representation spaces of genus zero Fuchsian groups. Math. Ann. 290(1991), 509526.Google Scholar
[4] Bauer, S. and Okonek, C., The algebraic geometry of representation spaces associated to Seifert fibred homology 3-spheres. Math. Ann. 286(1990), 4576.Google Scholar
[5] Biquard, O., Fibrés paraboliques stables et connexions singuli`eres plates. Bull. Soc. Math. France 119(1991), 231257.Google Scholar
[6] Biquard, O., On parabolic bundles over a complex surface. London, J. Math. Soc. 53(1996), 302316.Google Scholar
[7] Biswas, I. and Raghavendra, N., Determinants of parabolic bundles on Riemann surfaces. Proc. Indian Acad. Sci. Math. (1) 103(1993), 4171.Google Scholar
[8] Boden, H. U., Representations of orbifold groups and parabolic bundles. Comment. Math. Helv. 66(1991), 389447.Google Scholar
[9] Dolgachev, I., Algebraic surfaces with pg = q = 0. In: Algebraic surfaces CIME 1977, Liguori, Napoli, 1981, 97–215.Google Scholar
[10] Donaldson, S. K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. LondonMath. Soc. 50(1985), 126.Google Scholar
[11] Donaldson, S. K. and Kronheimer, P. B., The Geometry of Four-Manifolds. Clarendon Press, Oxford, 1990.Google Scholar
[12] Furuta, M. and Steer, B., Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points. Adv. Math. 96(1992), 38102.Google Scholar
[13] Griffith, P. and Harris, J., Principles of Algebraic Geometry. John Wiley & Sons, New York, 1978.Google Scholar
[14] Gantz, C., On parabolic bundles. D.Phil. thesis, Oxford University, 1996.Google Scholar
[15] Konno, H., On the natural line bundle on the moduli space of stable parabolic bundles. Commun.Math. Phys. 155(1993), 311324.Google Scholar
[16] Kronheimer, P. B. and Mrowka, T. S., Gauge theory for embedded surfaces, I. Topology 32(1993), 773826.Google Scholar
[17] Kronheimer, P. B. and Mrowka, T. S., Gauge theory for embedded surfaces, II. Topology 34(1995), 3798.Google Scholar
[18] Lübke, M., Stability of Einstein-Hermitian vector bundles. Manuscripta Math. 42(1983), 245257.Google Scholar
[19] Mehta, V. B. and Seshadri, C. S., Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248(1980), 205239.Google Scholar
[20] Miyaoka, Y., Kähler metrics on elliptic surfaces. Proc. Japan Acad. 50(1974), 533536.Google Scholar
[21] Moishezon, B., Complex surfaces and connected sums of complex projective planes. Lecture Notes in Math. 603, Springer, Berlin, 1977.Google Scholar
[22] Munari, A., Singular instantons and parabolic bundles over complex surfaces. D.Phil. thesis, Oxford University, 1993.Google Scholar
[23] Narasimhan, M. S. and Seshadri, C. S., Stable and unitary vector bundles on compact Riemann surfaces. Ann. Math. 82(1963), 540567.Google Scholar
[24] Nasatyr, E. B. and Steer, B., The Narasimhan-Seshadri theorem for parabolic bundles: an orbifold approach. Philos. Trans. Roy. Soc. London A 353(1995), 137171.Google Scholar
[25] Poritz, J. A., Parabolic vector bundles and Hermitian-Yang-Mills connections over a Riemann surface. Internat. Math, J.. 4(1993), 467501.Google Scholar
[26] Steer, B. and Wren, A., The Donaldson-Hitchin-Kobayashi correspondence for parabolic bundles over orbifold surfaces. Preprint, Oxford, 1992.Google Scholar
[27] Ue, M., On the diffeomorphism types of elliptic surfaces with multiple fibres. Invent.Math. 84(1986), 633643.Google Scholar