Arrangements of Lines and Algebraic Surfaces

  • F. Hirzebruch
Part of the Progress in Mathematics book series (PM, volume 36)


In recent years the Chern numbers c 1 2 and c 2 of algebraic surfaces have aroused special interest. For a minimal surface of general type they are positive and satisfy the inequality c 1 2 ≤ 3c 2 (see Miyaoka [22] anal Yau [31]) where the equality sign holds if and only if the universal cover of the surface is the unit ball (see Yau [31] and [23] §2 for the difficult and [11] for the easy direction of this equivalence). It is interesting to know which positive rational numbers ≤ 3 occur as c 1 2/c 2 for a minimal surface of general type. For a long time (before 1955) it was believed that c 1 2/c 2 ≤ 2, in other words that the signature of a surface of general type is nonpositive. It is interesting to find surfaces with 2 < c 1 2/c 2 ≤ 3. Some were found by Kodaira [19]. Other authors constructed more surfaces of this kind (Holzapfel [13], [14], Inoue [16], Livné [21], Mostow-Siu [25], Miyaoka [23]). Recently Mostow [24] and Deligne-Mostow [5] used a paper of E. Picard of 1885 to construct discrete groups of automorphisms of the unit ball leading to interesting surfaces. In fact, the author was stimulated by Mostow’s lecture at the Arbeitstagung 1981 to study the surface Y 1 (see §3.2) which is related to one of the surfaces of Mostow.


Minimal Surface General Type Galois Group Euler Number Algebraic Surface 
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  1. [1]
    Brieskorn, E.: Sur les groupes de tresses (d’après V.I. Arnold), Séminaire Bourbaki, 24e année, 1971/72, Lecture Notes 317, p. 21–44, Springer 1973.Google Scholar
  2. [2]
    Burr, St., Griinbaum, B., and Sloane, N. J. A.: The orchard problem, Geometriae Dedicata 2, p. 397–424 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Cartier, P.: Les arrangement d’hyperplans: Un chapitre de géométrie combinatoire, Séminaire Bourbaki, 33e année, 1980/81, Lecture Notes 901, p. 1–22, Springer 1981.Google Scholar
  4. [4]
    Deligne, P.: Les immeubles des groupes de tresses généralisés, Inv. Math. 17, p. 273–302 (1972).MathSciNetzbMATHGoogle Scholar
  5. [5]
    Deligne, P., and Mostow, G.D.: On a construction of Picard.Google Scholar
  6. [6]
    Fricke, R. und Klein, F.: Vorlesungen über die Theorie der automorphen Funktionen, zwei Bände, Nachdruck der 1. Auflage von 1897, Teubner Stuttgart 1965.Google Scholar
  7. [7]
    Grünbaum, B.: Arrangements of hyperplanes, in Proc. Second Louisiana Conference on Combinatorics and Graph Theory, p. 41106, Baton Rouge 1971.Google Scholar
  8. [8]
    Grünbaum, B.: Arrangements and spreads, Regional Conference Series in Mathematics, Number 10, Amer. Math. Soc. 1972.Google Scholar
  9. [9]
    Hansen, St.: Contributions to the Sylvester-Gallai-Theory, Polyteknisk Forlag Kobenhavn 1981.Google Scholar
  10. [10]
    Hirzebruch, F.: Der Satz von Riemann-Roch in Faisceau-theoretischer Formulierung, Proc. Intern. Congress Math. 1954, Vol. III, p. 457–473, North-Holland Publishing Company, Amsterdam 1956.Google Scholar
  11. [11]
    Hirzebruch, F.: Automorphe Formen und der Satz von Riemann-Roch, Symp. Intern. Top. Alg. 1956, p. 129–144, Universidad de México 1958.Google Scholar
  12. [12]
    Hirzebruch, F.: Some examples of algebraic surfaces, Proc. 21st Summer Research Institute Australian Math. Soc. 1981, p. 55–71, Contemporary Mathematics Vol. 9, Amer. Math. Soc. 1982.Google Scholar
  13. [13]
    Holzapfel, R.-P.: A class of minimal surfaces in the unknown region of surface geography, Math. Nachr. 98, p. 211–232 (1980).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Holzapfel, R.-P.: Invariants of arithmetic ball quotient surfaces, Math. Nachr. 103, p. 117–153 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Iitaka, S.: Geometry on complements of lines in P 2, Tokyo J. Math. 1, p. 1–19 (1978).MathSciNetzbMATHGoogle Scholar
  16. [16]
    Inoue, M.: Some surfaces of general type with positive indices, preprint 1981.Google Scholar
  17. [17]
    Ishida, M.-N.: The irregularities of Hirzebruch’s examples of sur- faces of general type with ci = 3c2, Math. Ann. (to appear).Google Scholar
  18. [18]
    Ishida, M.-N.: IIirzebruch’s examples of surfaces of general type with ci = 3c2, Japan-France Seminar 1982.Google Scholar
  19. [19]
    Kodaira, K.: A certain type of irregular algebraic surfaces, J. Analyse Math. 19, p. 207–215 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Kodaira, K.: On the structure of complex analytic surface W, Amer. J. of Math. 90, p. 1048–1066 (1967).MathSciNetCrossRefGoogle Scholar
  21. [21]
    Livné, R. A.: On certain covers of the universal elliptic curve, Ph. D. thesis, Harvard University 1981.Google Scholar
  22. [22]
    Miyaoka, Y.: On the Chern numbers of surfaces of general type, Inv. Math. 42, p. 225–237 (1977).MathSciNetzbMATHGoogle Scholar
  23. [23]
    Miyaoka, Y.: On algebraic surfaces with positive index, preprint 1980, SFB Theor. Math. Bonn.Google Scholar
  24. [24]
    Mostow, G.D.: Complex reflection groups and non-arithmetic monodromy, lecture at the Bonn Arbeitstagung June 1981, see also Proc. Nat. Acad. Sci. USA (to appear).Google Scholar
  25. [25]
    Mostow, G.D., and Siu, Y.-T.: A compact Kähler surface of nega- tive curvature not covered by the ball, Ann. of Math. 112, p. 321360 (1980).Google Scholar
  26. [26]
    Orlik, P., and Solomon, L.: Combinatorics and topology of comple- ments of hyperplanes, Inv. Math. 56, p. 167–189 (1980).MathSciNetzbMATHGoogle Scholar
  27. [27]
    Orlik, P., and Solomon, L.: Coxeter arrangements, Amer. Math. Soc. Proc. Symp. Pure Math 40 (to appear).Google Scholar
  28. [28]
    Orlik, P., and Solomon, L.: Arrangements defined by unitary reflec- tion groups, Math. Ann. (to appear).Google Scholar
  29. [29]
    Persson, U.: Chern invariants of surfaces of general type, Comp. Math. 43, p. 3–58 (1981).MathSciNetzbMATHGoogle Scholar
  30. [30]
    Shephard, G. C., and Todd, J. A.: Finite unitary reflection groups, Can. J. Math. 6, p. 274–304 (1954).MathSciNetzbMATHGoogle Scholar
  31. [31]
    Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74, p. 1798–1799 (1977).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • F. Hirzebruch
    • 1
  1. 1.Max-Planck-Institut für MathematikBonn 3Federal Republic of Germany

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