Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-03T00:19:57.921Z Has data issue: false hasContentIssue false
  • English
  • Français

The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces

Published online by Cambridge University Press:  20 November 2018

Brian Steer  and
Andrew Wren
Brian Steer
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK
Andrew Wren
Affiliation:
Mathematical Institute, 24–29 St. Giles, 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.

A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact Kähler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

Keywords

14J1757R57
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 6 , 01 December 2001 , pp. 1309 - 1339
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Ahlfors, L. V., Lectures on quasi-conformal mappings. Van Nostrand, Princeton, 1966.Google Scholar
[2] Atiyah, M. F. and Bott, R., The Yang Mills equations over Riemann Surfaces. Philos. Trans. Royal. Soc. London A308(1982), 5–3–615.Google Scholar
[3] Baily, W. L. Jr, On the imbedding of V-manifolds in projective space. Amer. J. Math. 79(1957), 403–430.Google Scholar
[4] Barth, W., Peters, C. and van de Ven, A., Compact complex surfaces. Springer-Verlag, Berlin, 1984.Google Scholar
[5] Biquard, O., Fibrés paraboliques stables et connexions singulières plates. Bull. Soc. Math. France 119(1991), 231–257.Google Scholar
[6] Biquard, O., On parabolic bundles over a complex surface. J. London Math. Soc. 53(1996), 302–316.Google Scholar
[7] Chevalley, C., Invariants of finite groups generated by reflections. Amer. J. Math. 77(1955), 778–782.Google Scholar
[8] Donaldson, S. K., Anti-self-dual Yang-Mills connexions over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3) 50(1985), 1–26.Google Scholar
[9] Donaldson, S. K., Boundary value broblems for Yang-Mills fields. J. Geom. Phys. 8(1992), 89–122.Google Scholar
[10] Donaldson, S. K., Furuta, M. and Kotschick, D., Floer homology groups in Yang-Mills theory. In preparation.Google Scholar
[11] Donaldson, S. K. and Kronheimer, P. B., The geometry of 4-manifolds. Oxford University Press, 1990.Google Scholar
[12] Furuta, M. and Steer, B., Seifert fibred homology 3-spheres and the Yang-Mills equation on Riemann surfaces with marked points. Adv. Math. 96(1992), 38–102.Google Scholar
[13] Griffiths, Philip and Harris, Joseph, Principles of Algebraic Geometry. Wiley, New York, 1978.Google Scholar
[14] Hamilton, R. S., Harmonic maps of manifolds with boundary. Lecture Notes in Math. 471, Springer-Verlag, Berlin, 1975.Google Scholar
[15] Hörmander, L., Linear partial differential operators. Springer-Verlag, New York-Berlin, 1976.Google Scholar
[16] Kawasaki, T., The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math. 16(1979), 151–159.Google Scholar
[17] Kronheimer, P. B. and Mrowka, T. S., Gauge theory for embedded surfaces I. Topology 32(1993), 773–826.Google Scholar
[18] Kronheimer, P. B. and Mrowka, T. S., Gauge theory for embedded surfaces II. Topology 34(1995), 37–97.Google Scholar
[19] Lockhart, R. B. and McOwen, R. C., Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(1985), 409–447.Google Scholar
[20] Looijenga, E. J. N., Isolated singular points on complete intersections. London Math. Soc. Lecture Note Ser. 77, Cambridge University Press, 1984.Google Scholar
[21] Maruyama, M. and Yokogawa, K., Moduli of parabolic stable sheaves. Math. Ann. 293(1992), 77–99.Google Scholar
[22] Mehta, V. B. and Seshadri, C. S., Moduli of vector bundles on curves with parabolic structure. Ann. Math. 248(1980), 205–239.Google Scholar
[23] Melrose, R. B., Pseudodifferential operators, corners and singular limits. In: Proc. International Congress of Mathematicians, Vols. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 217–234.Google Scholar
[24] Mumford, D., Lectures on curves on an algebraic surface. Ann. of Math. Stud. 59, Princeton Univ. Press, Princeton, 1966.Google Scholar
[25] Munari, A., Singular instantons and parabolic bundles over complex surfaces. D.Phil. thesis, Oxford, 1993.Google Scholar
[26] Narasimhan, M. S. and Seshadri, C. S., Stable and unitary vector bundles on compact Riemann surfaces. Ann.Math. 82(1965), 540–567.Google Scholar
[27] Nasatyr, E. B. and Steer, B., Orbifold Riemann Surfaces and the Yang-Mills-Higgs equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21(1995), 595–643.Google Scholar
[28] Nasatyr, E. B. and Steer, B., The Narasimhan-Seshadri theorem for parabolic bundles with rational weights: an orbifold approach. Philos. Trans. Roy. Soc. London A353(1995), 137–171.Google Scholar
[29] Râde, J., Singular Yang-Mills fields; local theory I. J. Reine Angew. Math. 452(1994), 111–151.Google Scholar
[30] Satake, I., On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. U.S.A. 42(1956), 359–363.Google Scholar
[31] Simpson, C. T., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1(1988), 867–918.Google Scholar
[32] Steer, B. and Wren, A., Grothendieck Topology and the Picard group of a complex orbifold. Contemp. Math. 239(1999), 251–262.Google Scholar
[33] Thurston, W. P., The geometry and topology of 3-manifolds. Princeton University Press, 1996.Google Scholar
[34] Tsuji, H., Stability of tangent bundles of minimal algebraic varieties. Topology 27(1988), 429–442.Google Scholar
[35] Uhlenbeck, K. K., Connexions with Lp bounds on curvature. Comm. Math. Phys. 83(1982), 31–34.Google Scholar
[36] Uhlenbeck, K. K., Removable singularities in Yang-Mills fields. Comm. Math. Phys. 83(1982), 11–30.Google Scholar
[37] Uhlenbeck, K. K. and Yau, S. T., On the existence of Yang-Mills connexions on stable bundles over compact Kähler manifolds. Comm. Pure Appl. Math. 39(1986), 257–293. (Correction: ibid 42, 703–707.)Google Scholar
[38] Wren, A. J., The geometry of complex orbifolds. D.Phil thesis, Oxford, 1993.Google Scholar