Advertisement

Stable Bundles, Instantons and C∞-Structures on Algebraic Surfaces

  • Ch. Okonek
  • A. Van de Ven

Abstract

The ties between topology and algebraic geometry go back to the nineteenth century, but specific questions about surfaces being homeomorphic or not, have only come up in the fifties and sixties. Seven asked in 1954 if every algebraic surface homeomorphic to the projective plane, is biregularly equivalent to it. The (positive) answer came three decades later, as a result of Yau’s work on the Calabi conjecture. In 1965 Kodaira constructed examples of elliptic surfaces with the homotopy type of a K3-surface, and he posed the question, whether they are homeomorphic to such a surface. At the time, the only theorem known was a result of J.H.C. Whitehead, stating that two simply-connected surfaces are of the same homotopy type if and only if they have isomorphic (integer-valued) intersections forms. Given the known structure theorems for indefinite forms, it was a simple matter to verify this in Kodaira’s case. The affirmative answer to Kodaira’s question and many similar ones is an immediate consequence of Freedman’s fundamental theorem from 1982, saying in particular that the homeomorphism type of a compact, oriented, simply-connected differentiable 4-fold is completely determined by its intersection form.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A/K]
    Altman, A.B., Kleiman, S.L.: Compactifying the Picard scheme. Adv. Math. 35 (1980) 50–112zbMATHMathSciNetGoogle Scholar
  2. [A/D/H/M]
    Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Y.I.: Constructions of instantons. Phys. Lett. 65A (1978) 185–187MathSciNetGoogle Scholar
  3. [A/H/S]
    Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self duality in four dimensional Riemannian geometry. Proc. Roy. Soc. London A 362 (1978) 425–461MathSciNetGoogle Scholar
  4. [B]
    Bănică, C: Topologisch triviale holomorphe Vektorbündel auf n. Crelle’s J. 344 (1983) 102–119zbMATHGoogle Scholar
  5. [B/L]
    Bănică, C., Le Potier, J.: Sur l’existence des fibrés vectoriels holomorphes sur les surfaces non-algébriques. J. reine angew. Math. 378 (1987) 1–31zbMATHMathSciNetGoogle Scholar
  6. [BA]
    Barlow, R.: A simply connected surface of general type with p g = 0. Invent. math. 79 (1985) 293–301zbMATHMathSciNetGoogle Scholar
  7. [BAR]
    Barth, W.: Moduli of vector bundles on the projective plane. Invent. Math. 42 (1977) 63–91zbMATHMathSciNetGoogle Scholar
  8. [B/P/V]
    Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4.) Springer, Berlin Heidelberg New York Tokyo 1984zbMATHGoogle Scholar
  9. [B/O]
    Bauer, S., Okonek, C.: The algebraic geometry of representation spaces associated to Seifert fibered homology 3-spheres. Preprint 27, MPI Bonn 1989Google Scholar
  10. [BE]
    Besse, A.: Géométrie riemannienne en dimension 4. Séminaire Arthur Besse 1978/79. “Textes mathématiques” 3. Cedic, Paris 1981Google Scholar
  11. [BOU]
    Bourguignon, J.-P: Analytical problems arising in geometry: examples from Yang-Mills theory. Jahresber. d. Dt. Math. Verein. 87 (1985) 67–89zbMATHMathSciNetGoogle Scholar
  12. [BR 1]
    Brosius, J.E.: Rank-2 vector bundles on a ruled surface I. Math. Ann. 265 (1983) 155–168zbMATHMathSciNetGoogle Scholar
  13. [BR 2]
    Brosius, J.E.: Rank-2 vector bundles on a ruled surface II. Math. Ann. 266 (1983) 199–214zbMATHMathSciNetGoogle Scholar
  14. [BRU]
    Brussee, R.: Stable bundles on blown up surfaces. Preprint, Leiden 1989Google Scholar
  15. [DO]
    Dolgachev, I.: Algebraic surfaces with q = pg = 0 In: Algebraic surfaces. C.I.M.E., Liguori Napoli (1981), pp. 97–215Google Scholar
  16. [D1]
    Donaldson, S.K.: A new proof of a theorem of Narasimhan and Seshadri. J. Diff. Geom. 18 (1983) 269–277zbMATHMathSciNetGoogle Scholar
  17. [D2]
    Donaldson, S.K.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. 50 (1985) 1–26zbMATHMathSciNetGoogle Scholar
  18. [D3]
    Donaldson, S.K.: Connections, cohomology and the intersection forms of 4-manifolds. J. Diff. Geom. 24 (1986) 275–341zbMATHMathSciNetGoogle Scholar
  19. [D 4]
    Donaldson, S.K.: Irrationality and the h-cobordism conjecture. J. Diff. Geom. 26 (1987) 141–168zbMATHMathSciNetGoogle Scholar
  20. [D5]
    Donaldson, S.K.: Polynomial invariants for smooth 4-manifolds. Topology 29, no. 3 (1990) 257–315zbMATHMathSciNetGoogle Scholar
  21. [D6]
    Donaldson, S.K.: The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Diff. Geom. (to appear)Google Scholar
  22. [D7]
    Donaldson, S.K.: Letter to Okonek, December 1988Google Scholar
  23. [D8]
    Donaldson, S.K.: Talk at Durham, July 1989Google Scholar
  24. [DR 1]
    Drezet, J.: Fibrés exceptionnels et variétés de modules de faisceaux semistables sur ℙ2 Crelle’s J. 380 (1987) 14–58zbMATHMathSciNetGoogle Scholar
  25. [DR2]
    Drezet, J.: Cohomologie des variétés de modules de hauteur nulle. Math. Ann. 281 (1988) 43–85zbMATHMathSciNetGoogle Scholar
  26. [DR 3]
    Drezet, J.: Groupe de Picard des variétés de modules de faisceaux semi-stables sur ℙ2(C). Ann. Inst. Fourier, Grenoble 38 (3) (1988) 105–168zbMATHMathSciNetGoogle Scholar
  27. [E 1]
    Ebeling, W.: An arithmetic characterization of the symmetric monodromy groups of singularities. Invent. math. 77 (1984) 85–99zbMATHMathSciNetGoogle Scholar
  28. [E 2]
    Ebeling, W.: An example of two homeomorphic, nondiffeomorphic complete intersection surfaces. Invent. math. 99 (1990) 651–654zbMATHMathSciNetGoogle Scholar
  29. [E/F]
    Elencwajg, G., Forster, O.: Vector bundles on manifolds without divisors and a theorem on deformations. Ann. Inst. Fourier, Grenoble 32 (1982) 25–51zbMATHMathSciNetGoogle Scholar
  30. [F/S]
    Fintushel, R, Stern, R: SO(3)-connections and the topology of 4-manifolds. J. Diff. Geom. 20 (1984) 523–539zbMATHMathSciNetGoogle Scholar
  31. [F/U]
    Freed, D., Uhlenbeck, K.K.: Instantons and four manifolds. M.S.R.I. publ. no. 1. Springer, New York Berlin Heidelberg Tokyo 1984zbMATHGoogle Scholar
  32. [F]
    Freedman, M.: The topology of 4-manifolds. J. Diff. Geom. 17 (1982) 357–454zbMATHGoogle Scholar
  33. [FR]
    Friedman, R: Rank two vector bundles over regular elliptic surfaces. Invent. math. 96 (1989) 283–332zbMATHMathSciNetGoogle Scholar
  34. [F/M/M]
    Friedman, R, Moishezon, B., Morgan, J.W.: On the C invariance of the canonical classes of certain algebraic surfaces. Bull. A.M.S. 17 (2) (1987) 283–286zbMATHMathSciNetGoogle Scholar
  35. [F/M 1]
    Friedman, R, Morgan, J.W.: Algebraic surfaces and 4-manifolds: some conjectures and speculations. Bull. A.M.S. 18 (1) (1988) 1–15zbMATHMathSciNetGoogle Scholar
  36. [F/M 2]
    Friedman, R, Morgan, J.W.: On the diffeomorphism type of certain algebraic surfaces I. J. Diff. Geom. 27 (1988) 297–369zbMATHMathSciNetGoogle Scholar
  37. [F/M 3]
    Friedman, R, Morgan, J.W.: On the diffeomorphism type of certain algebraic surfaces II. J. Diff. Geom. 27 (1988) 371–398zbMATHMathSciNetGoogle Scholar
  38. [F/M 4]
    Friedman, R, Morgan, J.W.: Complex versus differentiable classification of algebraic surfaces. Preprint, New York 1988Google Scholar
  39. [F/S]
    Fujiki, A., Schuhmacher, G.: The moduli space of Hermite-Einstein bundles on a compact Kähler manifold. Proc. Japan Acad. 63 (1987) 69–72zbMATHGoogle Scholar
  40. [G]
    Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math. 106 (1977) 45–60zbMATHMathSciNetGoogle Scholar
  41. [GO]
    Gompf, R.E.: On sums of algebraic surfaces. Preprint, Austin 1988Google Scholar
  42. [G/H]
    Griffiths, Ph., Harris, J.: Residues and zero-cycles on algebraic varieties. Ann. Math. 108 (1978) 461–505zbMATHMathSciNetGoogle Scholar
  43. [GR]
    Grothendieck, A.: Techniques de construction et théorèmes d’existence en géométrie algébrique, IV: Les schémas de Hilbert. Sém. Bourbaki no. 221 (1961)Google Scholar
  44. [H/H]
    Hirzebruch, F., Hopf, H.: Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten. Math. Ann. 136 (1956) 156–172MathSciNetGoogle Scholar
  45. [H/S]
    Hoppe, H.J., Spindler, H.: Modulräume stabile 2-Bündel auf Regelflächen. Math. Ann. 249 (1980) 127–140zbMATHMathSciNetGoogle Scholar
  46. [H]
    Hulek, K.: Stable rank-2 vector bundles on ℙ2 with c1 odd. Math. Ann. 242 (1979) 241–266MathSciNetGoogle Scholar
  47. [K1]
    Kobayashi, S.: First Chern class and holomorphic tensor fields. Nagoya Math. J. 77 (1980) 5–11zbMATHMathSciNetGoogle Scholar
  48. [K2]
    Kobayashi, S.: Differential geometry of complex vector bundles. Iwanami Shoten and Princeton University Press 1987zbMATHGoogle Scholar
  49. [K/O]
    Kosarew, S., Okonek, C.: Global moduli spaces and simple holomorphic bundles. Publ. R.I.M.S., Kyoto Univ. 25 (1989) 1–19Google Scholar
  50. [KOT]
    Kotschick, D.: On manifolds homeomorphic \([C(\mathbb{P})^2 \ne 8C\bar (\mathbb{P})^2 ]\). Invent. math. 95 (1989) 591–600zbMATHMathSciNetGoogle Scholar
  51. [L]
    Lawson, H.B.: The theory of gauge fields in four dimensions. Regional Conf. Series, A.M.S. 58. Providence, Rhode Island 1985Google Scholar
  52. [LP]
    Le Potier, J.: Fibres stables de rang 2 sur ℙ2(C). Math. Ann. 241 (1979) 217–256zbMATHMathSciNetGoogle Scholar
  53. [LU 1]
    Lübke, M.: Chernklassen von Hermite-Einstein Vektorbündeln. Math. Ann. 260 (1982) 133–141zbMATHMathSciNetGoogle Scholar
  54. [LU/2]
    Lübke, M.: Stability of Einstein-Hermitian vector bundles. Manuscr. Math. 42 (1983) 245–257zbMATHGoogle Scholar
  55. [L/Ol]
    Lübke, M., Okonek, C: Moduli spaces of simple bundles and Hermitian-Einstein connections. Math. Ann. 267 (1987) 663–674Google Scholar
  56. [L/02]
    Lübke, M., Okonek, C.: Stable bundles on regular elliptic surfaces. Crelle’s J. 378 (1987) 32–45zbMATHGoogle Scholar
  57. [MA]
    Mandelbaum, R: Four-dimensional topology: an introduction. Bull. A.M.S. 2 (1) (1980) 1–159zbMATHMathSciNetGoogle Scholar
  58. [MAG]
    Margerin, C.: Fibrés stables et metriques d’Hermite-Einstein. Sém. Bourbaki no. 683 (1987)Google Scholar
  59. [M 1]
    Maruyama, M.: Stable bundles on an algebraic surface. Nagoya Math. J. 58 (1975) 25–68zbMATHMathSciNetGoogle Scholar
  60. [M 2]
    Maruyama, M.: Moduli of stable sheaves I. J. Math. Kyoto Univ. 17 (1977) 91–126zbMATHMathSciNetGoogle Scholar
  61. [M 3]
    Maruyama, M.: Moduli of stable sheaves II. J. Math. Kyoto Univ. 18 (1978) 557–614zbMATHMathSciNetGoogle Scholar
  62. [M 4]
    Maruyama, M.: Elementary transformations in the theory of algebraic vector bundles. In: Aroca, J.M., Buchweitz, R, Giusti, M., Merle, M. (eds.): Algebraic Geometry, Proc. La Rábida (Lecture Notes in Mathematics, vol.961). Springer, Berlin Heidelberg New York 1982, pp. 241–266Google Scholar
  63. [M 5]
    Maruyama, M.: Moduli of stable sheaves – generalities and the curve of jumping lines of vector bundles on ℙ2. Advanced Studies of Pure Math., I, Alg. Var. and Anal. Var. 1–27, Kinokuniya and North-Holland 1983Google Scholar
  64. [M 6]
    Maruyama, M.: The equations of plane curves and the moduli spaces of vector bundles on ℙ2. In: Algebraic and topological theories, to the memory of T. Miyata, Tokyo 1985, pp. 430–466Google Scholar
  65. [M 7]
    Maruyama, M.: Vector bundles on ℙ2 and torsion sheaves on the dual plane. In: Vector bundles on Algebraic Varieties. Proc. Bombay 1984. Oxford Univ. Press 1987, pp. 275–339Google Scholar
  66. [MI]
    Miyajima, K.: Kuranshi family of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections. Publ. R.I.M.S., vol. 25, Kyoto Univ., 1989, pp. 301–320Google Scholar
  67. [MO 1]
    Mong, K.-C.: Some polynomials on \([(\mathbb{P})_2 (\mathbb{C}) \ne \bar (\mathbb{P})_2 (\mathbb{C})]\). Preprint 32, M.P.I. Bonn 1989Google Scholar
  68. [MO 2]
    Mong, K.-C.: On some possible formulation of differential invariants for 4-manifolds. Preprint 34, M.P.I. Bonn 1989Google Scholar
  69. [MO 3]
    Mong, K.-C.: Moduli spaces of stable 2-bundles and polarizations. Preprint 36, M.P.I. Bonn 1989Google Scholar
  70. [MO 4]
    Mong, K.-C.: Polynomial invariants for 4-manifolds of type (l,n) and a calculation for S2 x S2. Preprint 37, M.P.I. Bonn 1989Google Scholar
  71. [MU]
    Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. math. 77 (1984) 101–116zbMATHMathSciNetGoogle Scholar
  72. [N/S]
    Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on compact Riemann surfaces. Ann. Math. 82 (1965) 540–567zbMATHMathSciNetGoogle Scholar
  73. [N]
    Norton, V.A.: Analytic moduli of complex vector bundles. Indiana Univ. Math. J. 28 (1979) 365–387zbMATHMathSciNetGoogle Scholar
  74. [O]
    Okonek, C.: Fake Enriques surfaces. Topology 23 (4) (1988) 415–427MathSciNetGoogle Scholar
  75. [O/S/S]
    Okonek, C., Schneider, M., Spindler, H.: Vector bundles over complex projective spaces. Progress in Math. 3. Birkhäuser, Boston Basel Stuttgart 1980Google Scholar
  76. [O/V1]
    Okonek, C., Van de Ven, A.: Stable bundles and differentiable structures on certain elliptic surfaces. Invent. math. 86 (1986) 357–370zbMATHMathSciNetGoogle Scholar
  77. [O/V 2]
    Okonek, C., Van de Ven, A.: Г-type-invariants associated to PU(2)-bundles and the differentiable structure of Barlow’s surface. Invent. math. 95 (1989) 601–614zbMATHMathSciNetGoogle Scholar
  78. [P]
    Peters, C.A.M.: On two types of surfaces of general type with vanishing geometric genus. Invent. math. 32 (1976) 33–47zbMATHMathSciNetGoogle Scholar
  79. [SA]
    Salvetti, M.: On the number of non-equivalent differentiable structures on 4-manifolds. Manuscr. Math. 63 (1989) 157–171zbMATHMathSciNetGoogle Scholar
  80. [S 1]
    Schwarzenberger, R.L.E.: Vector bundles on algebraic surfaces. Proc. London Math. Soc. (3) 11 (1961) 601–623zbMATHMathSciNetGoogle Scholar
  81. [S 2]
    Schwarzenberger, RL.E.: Vector bundles on the projective plane. Proc. London Math. Soc. (3) 11 (1961) 623–640zbMATHMathSciNetGoogle Scholar
  82. [SE]
    Sedlacek, S.: A direct method for minimizing the Yang-Mills functional over 4-manifolds. Commun. Math. Phys. 86 (1982) 515–528zbMATHMathSciNetGoogle Scholar
  83. [SER]
    Serre, J.-P.: A course in arithmetic. (Graduate Texts in Mathematics, vol.7). Springer, New York Heidelberg Berlin 1973zbMATHGoogle Scholar
  84. [SM]
    Smale, S.: Generalized Poincaré’s conjecture in dimensions > 4. Ann. Math. 74 (1961) 391–466zbMATHMathSciNetGoogle Scholar
  85. [SP]
    Spanier, E.H.: Algebraic topology. McGraw-Hill 1966zbMATHGoogle Scholar
  86. [ST1]
    Strømme, S.A.: Deforming vector bundles on the projective plane. Math. Ann. 263 (1983) 385–397MathSciNetGoogle Scholar
  87. [ST 2]
    Strømme, S.A.: Ample divisors on fine moduli spaces on the projective plane. Math. Z. 187 (1984) 405–423MathSciNetGoogle Scholar
  88. [T]
    Takemoto, F.: Stable vector bundles on algebraic surfaces. Nagoya Math. J. 47 (1972) 29–48zbMATHMathSciNetGoogle Scholar
  89. [TA]
    Taubes, C.H.: Self-dual connections on 4-manifolds with indefinite intersection matrix. J. Diff. Geom. 19 (1984) 517–560zbMATHMathSciNetGoogle Scholar
  90. [UE]
    Ue, M.: On the diffeomorphism types of elliptic surfaces with multiple fibres. Invent. math. 84 (1986) 633–643zbMATHMathSciNetGoogle Scholar
  91. [U 1]
    Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys. 83 (1982) 11–30zbMATHMathSciNetGoogle Scholar
  92. [U 2]
    Uhlenbeck, K.K.: Connections with L P bounds on curvature. Commun. Math. Phys. 83 (1982) 31–42zbMATHMathSciNetGoogle Scholar
  93. [U/Y]
    Uhlenbeck, K.K., Yau, S.-T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39 (1986) 257–293MathSciNetGoogle Scholar
  94. [UM]
    Umemura, H.: Stable vector bundles with numerically trivial chern classes over a hyperelliptic surface. Nagoya Math. J. 59 (1975) 107–134zbMATHMathSciNetGoogle Scholar
  95. [V 1]
    Van de Ven, A.: Twenty years of classifying algebraic vector bundles. In: Journées de géométrie algébrique. Sijthoff and Noordhoff, Alphen aan den Rijn 1980Google Scholar
  96. [V 2]
    Van de Ven, A.: On the differentiable structure of certain algebraic surfaces. Sém. Bourbaki no. 667 (1986)Google Scholar
  97. [W 1]
    Wall, C.T.C.: Diffeomorphisms of 4-manifolds. J. London Math. Soc. 39 (1964) 131–140zbMATHMathSciNetGoogle Scholar
  98. [W 2]
    Wall, C.T.C.: On simply-connected 4-manifolds. J. London Math. Soc. 39 (1964) 141–149zbMATHMathSciNetGoogle Scholar
  99. [WE]
    Wehler, J.: Moduli space and versai deformation of stable vector bundles. Rev. Roumaine Math. Pures Appl. 30 (1985) 69–78zbMATHMathSciNetGoogle Scholar
  100. [WEL]
    Wells, R.O.: Differential analysis on complex manifolds. (Graduate Texts in Mathematics, vol. 65). Springer, New York Heidelberg Berlin 1980zbMATHGoogle Scholar
  101. [WU]
    Wu, W.: Sur les espaces fibrés. Publ. Inst. Univ. Strasbourg, XI. Paris 1952zbMATHGoogle Scholar
  102. [Y]
    Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74 (1977) 1789–1799Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Ch. Okonek
  • A. Van de Ven

There are no affiliations available

Personalised recommendations