×

zbMATH — the first resource for mathematics

Parabolic bundles, elliptic surfaces and \(SU(2)\)-representation spaces of genus zero Fuchsian groups. (English) Zbl 0752.14035
The paper considers representation of Fuchsian groups of finite type with fixed weight at the parabolic elements within the concept of parabolic bundles introduced by Seshadri. The main aim of the paper is to give the explicit description of moduli spaces of parabolic bundles in the case of rank two bundles over the projective line. The algebro-geometric construction of the isomorphism of parabolic push arounds with the moduli space is used.

MSC:
14H60 Vector bundles on curves and their moduli
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H55 Riemann surfaces; Weierstrass points; gap sequences
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. (Erg. Math. Grenzgeb. Bd. 3, 4). Berlin Heidelberg New York: Springer 1985
[2] Bauer, S., Okonek, C.: The algebraic geometry of representation spaces associated to Seifert fibered homology 3-spheres. Math. Ann.286, 45-76 (1990) · Zbl 0723.57012 · doi:10.1007/BF01453565
[3] Dolgachev, I.: Algebraic surfaces withp g =q=0. In: Algebraic surfaces CIME 1977, Liguori, Napoli, 97-215 (1981)
[4] Donaldson, S.K.: Anti-selfdual Yang-Mills connections over an algebraic surface and stable vector bundles. Proc. Lond. Math. Soc.50, 1-26 (1985) · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1
[5] Fintushel, R., Stern, R.: Instanton homology of Seifert fibered homology three spheres. Proc. Lond. Math. Soc.61, 109-137 (1990) · Zbl 0705.57009 · doi:10.1112/plms/s3-61.1.109
[6] Forster, O., Knorr, K.: ?ber die Deformationen von Vektorraumb?ndeln auf kompakten komplexen R?umen. Math. Ann.209, 291-346 (1974) · Zbl 0281.32015 · doi:10.1007/BF01351725
[7] Furuta, M., Steer, B.: The moduli spaces of flat connections on certain 3-manifolds. To appear in Adv. Math. · Zbl 0769.58009
[8] Grothendieck, A.: El?ments de g?ometrie alg?braique. EGA III, Publ. Math. Inst. Hautes Etud. Sci.11 (1961)
[9] Kirk, P., Klassen, E.: Representation Spaces of Seifert fibered homology spheres. Topology30, 77-95 (1991) · Zbl 0721.57007 · doi:10.1016/0040-9383(91)90035-3
[10] Kobayashi, S.: Differential geometry of complex vector bundles. Princeton: 1987 · Zbl 0708.53002
[11] L?bke, M.: Chernklassen von Hermite-Einstein-Vektorb?ndeln. Math. Ann.260, 133-141 (1982) · Zbl 0481.53058 · doi:10.1007/BF01475761
[12] Maruyama, M.: Moduli of stable sheaves. I, II. J. Math. Kyoto Univ.17, 91-126 (1977);18, 557-614 (1978) · Zbl 0374.14002
[13] Mehta, V., Seshadri, C.S.: Moduli of vector bundles on curves with parabolic structures. Math. Ann.248, 205-239 (1980) · Zbl 0454.14006 · doi:10.1007/BF01420526
[14] Miyajima, K.: A note on moduli spaces of simple vector bundles. Publ. Res. Inst. Math. Sci. Kyoto Univ.25, 301-320 (1989) · Zbl 0683.32016 · doi:10.2977/prims/1195173613
[15] Miyaoka, Y.: K?hler metrics on elliptic surfaces. Proc. Japan Acad.50, 533-536 (1974) · Zbl 0354.32011 · doi:10.3792/pja/1195518827
[16] Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact surface. Ann. Math.82, 540-567 (1965) · Zbl 0171.04803 · doi:10.2307/1970710
[17] Nisture, N.: Cohomology of the moduli of parabolic vector bundles. Proc. Indian Acad. Sci. Math. Sci.95, 61-77 (1986) · Zbl 0611.14014 · doi:10.1007/BF02837250
[18] Seshadri, C.S.: Fibr?s vectoriels sur les courbes alg?briques. Asterisque96 (1982) · Zbl 0517.14008
[19] Seshadri, C.S.: Desingularisation of moduli varieties of vector bundles on curves. Proc. Kyoto conf. Alg. Geom. 155-184 (1977) · Zbl 0412.14005
[20] Smale, S.: Structure of manifolds. Am. J. Math.84, 387-399 (1962) · Zbl 0109.41103 · doi:10.2307/2372978
[21] Ue, M.: On the diffeomorphism types of elliptic surfaces with multiple fibers. Invent. Math.84, 633-643 (1986) · Zbl 0595.14028 · doi:10.1007/BF01388750
[22] Griffiths, R., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.