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The Theory of Teichmüller Spaces A View Towards Moduli Spaces of Kähler Manifolds

  • Georg Schumacher

Abstract

Over the last five decades, beautiful results have been proved in the subject of Teichmüller theory. Recently this area has been influenced by the spirit of analytic and algebraic geometry as well as complex differential geometry. Deformation theory of compact complex manifolds was created in a seemingly independent way. Its methods are significantly different and, as opposed to its classical counterpart, deformation theory only provides a local solution of the classification problem. A (coarse) moduli space, i.e. a global parameter space for complex structures exists only under certain assumptions. The aim of this article is to discuss some aspects of Teichmüller theory and their relationships to recent results on moduli of compact complex manifolds.

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References

  1. [AB 1980]
    Abikoff, W.: The real analytic theory of Teichmüller space. Lecture Notes in Mathematics, vol. 820) Springer, Berlin Heidelberg 1980zbMATHGoogle Scholar
  2. [AH 1938a]
    Ahlfors, L.V.: On quasiconformal mappings. Journal d’Analyse Math. 3 (1938) 359–364Google Scholar
  3. [AH 1938b]
    Ahlfors, L.V.: An extension of Schwarz’s lemma. Transactions of the AMS 43 (1938) 359–364MathSciNetGoogle Scholar
  4. [AH 1953]
    Ahlfors, L.V.: On quasiconformal mappings. J. d’Analyse Math. 3 (1953) 1–58CrossRefMathSciNetGoogle Scholar
  5. [AH 1960]
    Ahlfors, L.V.: The complex analytic structure of the space of closed Riemann surfaces. In: Analytic Functions. Princeton University Press 1960Google Scholar
  6. [AH 1961a]
    Ahlfors, L.V.: Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 74 (1961) 171–191CrossRefzbMATHMathSciNetGoogle Scholar
  7. [AH 1961b]
    Ahlfors, L.V.: Curvature properties of Teichmüller’s space. J. d’Analyse Math. 9 (1961) 161–176CrossRefzbMATHMathSciNetGoogle Scholar
  8. [A-B 1960]
    Ahlfors, L., Bers, L.: Riemann mapping theorem for variable metrics. Ann. Math. 72 (1960) 385–404CrossRefzbMATHMathSciNetGoogle Scholar
  9. [A-G 1962]
    Andreotti, A., Grauert, H.: Théorème de finitude pour la cohomologie des espace complexes. Bull. Soc. Math. France 90 (1962) 193–259zbMATHMathSciNetGoogle Scholar
  10. [BA 1957]
    Baily, W.L.: On the imbedding of V-manifolds in projective space. Am. J. Math. 79 (1957) 403–430CrossRefzbMATHMathSciNetGoogle Scholar
  11. [BA 1962]
    Baily, W.L.: On the theory of Θ-functions, the moduli of abelian varieties and the moduli space of curves. Ann. Math. 75 (1962) 342–381CrossRefzbMATHMathSciNetGoogle Scholar
  12. [B-B 1966]
    Baily, W.L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math 84 (1966) 442–528CrossRefzbMATHMathSciNetGoogle Scholar
  13. [B-E 1969]
    Berger, M., Ebin, D.G.: Some decompositions on the spaces of symmetric tensors on a Riemannian manifold. J. Difif. Geom. 3 (1969) 379–392zbMATHMathSciNetGoogle Scholar
  14. [BE 1960]
    Bers, L.: Spaces of Riemann surfaces. Proc. Int. Cong. 1958, Cambridge 1960Google Scholar
  15. [BE 1970]
    Bers, L.: On boundaries of Teichmüller spaces and Kleinian groups I. Ann. Math. 91 (1970) 570–600CrossRefzbMATHMathSciNetGoogle Scholar
  16. [BE 1974]
    Bers, L.: Spaces of degenerating Riemann surfaces, discontinous groups and Riemann surfaces. Princeton University Press, Princeton 1974Google Scholar
  17. [B-G-S 1987]
    Bismut, J.M., Gillet, H., Soulé, Ch.: Analytic torsion and holomorphic determinant bundles, I, II, III. Comm. Math. Phys. 115 (1987) 49–87, 79–126, 301–351CrossRefGoogle Scholar
  18. [CA 1960]
    Calabi, E.: On compact Riemann manifolds with constant curvature, I. AMS Proc. Symp. Pure Math. III (1960) 155–180Google Scholar
  19. [CA 1979]
    Calabi, E.: Extremal Kähler metrics. In: Yau, S.T. (ed.) Seminars on differential geometry. Princeton 1979Google Scholar
  20. [CA 1985]
    Calabi, E.: Extremal Kähler metrics II. In: Cheval, I., Farkas, H.M. (eds.) Differential geometry and complex analysis, dedicated to E. Rauch. Springer, Berlin Heidelberg 1985, pp. 259–290Google Scholar
  21. [C-S 1990]
    Campana, F., Schumacher, G.: A geometric algebraicity property for moduli spaces of compact Kähler manifolds with h 2,0= 1. Math. Z. 204 (1990) 153–155CrossRefzbMATHMathSciNetGoogle Scholar
  22. [DO 1987]
    Donaldson, S.K.: Infinite determinants, stable bundles and curvature. Duke Math. J. 54 (1987) 231–247CrossRefzbMATHMathSciNetGoogle Scholar
  23. [E-K 1974]
    Earle, C.J., Kra, I.: On holomorphic mappings between Teichmüller spaces. In: Ahlfors, L., Kra, I., Maskit, B., Nirenberg, L., (eds.). Contributions to Analysis. New York London 1974Google Scholar
  24. [F-N]
    Fenchel, W., Nielsen, J.: J. Discontinous groups of non-Euklidean motions. Unpublished manuscriptGoogle Scholar
  25. [F-T 1984a]
    Fischer, A.E., Tromba A.J.: On a purely Riemannian proof of the structure and dimension of the unramified moduli space of a compact Riemann surface. Math. Ann. 267 (1984) 311–345CrossRefzbMATHMathSciNetGoogle Scholar
  26. [F-T 1984b]
    Fischer, A.E., Tromba A.J.: On the Weil-Petersson metric on Teichmüller space. Trans. AMS 284 (1984) 311–345Google Scholar
  27. [F-T 1987]
    Fischer, A.E., Tromba, A.J.: A new proof that Teichmüller’s space is a cell. Trans. AMS 303 (1987) 257–262zbMATHMathSciNetGoogle Scholar
  28. [F-K 1972]
    Forster, O., Knorr, K.: Relativ-analytische Räume und die Kohärenz von Bildgarben. Invent. math. 16 (1972) 113–160CrossRefzbMATHMathSciNetGoogle Scholar
  29. [F-K 1926]
    Fricke, R, Klein, F.: Vorlesungen über die Theorie der automorphen Funktionen. Leipzig 1926Google Scholar
  30. [FU 1978]
    Fujiki, A.: On automorphism groups of compact Kähler manifolds. Invent. math 44 (1978) 226–258CrossRefMathSciNetGoogle Scholar
  31. [FU 1981]
    Fujiki, A.: A theorem on bimeromorphic maps of Kähler manifolds and its applications. Publ. RIMS Kyoto 17 (1981) 735–754Google Scholar
  32. [FU 1984]
    Fujiki, A.: Coarse moduli spaces for polarized Kähler manifolds. Publ. RIMS, Kyoto 20 (1984) 977–1005Google Scholar
  33. [F-S 1988a]
    Fujiki, A., Schumacher, G.: The moduli space of Kähler structures on a real symplectic manifold. Publ. RIMS, Kyoto 24 (1988) 141–168Google Scholar
  34. [F-S 1988b]
    Fujiki, A., Schumacher, G.: The moduli space of extremal, compact Kähler manifolds and generalized Weil-Petersson metrics. Preprint 1988. Publ. RIMS, Kyoto 26 (1990) 101–183Google Scholar
  35. [G-H 1988]
    Gerritsen, L., Herrlich, F.: The extended Schottky space. J. reine angew. Math. 389 (1988) 190–208MathSciNetGoogle Scholar
  36. [G-R 1954]
    Gerstenhaber, M., Rauch, H.E.: On extremal quasi-conformal mappings, I, II. Proc. Nat. Acad. Sci. 40 (1954) 808–812, 991–994CrossRefMathSciNetGoogle Scholar
  37. [GR 1928]
    Grötzsch, H.: Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes. Leipz. Ber. 80 (1928)Google Scholar
  38. [GR 1961]
    Grothendieck, A.: Technique de construction en géométrie analytique. Sém. Cartan no. 7–17 (1960/61)Google Scholar
  39. [HA 1983]
    Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. math. 72 (1983) 221–231CrossRefzbMATHMathSciNetGoogle Scholar
  40. [H-M 1982]
    Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. Invent. math. 67 (1982) 23–86CrossRefzbMATHMathSciNetGoogle Scholar
  41. [HA 1984]
    Harris, J.: On the Kodaira dimension of the moduli space of curves, II. The even-genus case. Invent. math. 75 (1984) 437–466CrossRefzbMATHMathSciNetGoogle Scholar
  42. [HE 1990]
    Herrlich, F.: The extended Teichmüller space. Math. Z. 203 (1990) 279–291CrossRefzbMATHMathSciNetGoogle Scholar
  43. [J-Y 1987]
    Jost, J., Yau, S.T.: On the rigidity of certain discrete groups and algebraic varieties. Math. Ann. 278 (1987) 481–496CrossRefzbMATHMathSciNetGoogle Scholar
  44. [JO 1991]
    Jost, J.: Harmonic maps and curvature computations in Teichmüller theory. Ann. Acad. Fenn. Ser. A 16 (1991) 13–46MathSciNetGoogle Scholar
  45. [KE 1971]
    Keen, L.: On Fricke moduli. Ann. Math. Studies 66 (1971) 205–224MathSciNetGoogle Scholar
  46. [K-M 1976]
    Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves, I: Preliminaries on “det” and “div”. Math. Scand. 39 (1976) 19–55zbMATHMathSciNetGoogle Scholar
  47. [KN 1983a]
    Knudsen, F.: The projectivity of the moduli space of stable curves, II: The stacks Mg,n. Math. Scand. 52 (1983) 161–199zbMATHMathSciNetGoogle Scholar
  48. [KN 1983b]
    Knudsen, F.: The projectivity of the moduli space of stable curves, III: The line bundles on M g,n and a proof of the projectivity of M g,n in characteristic 0. Math. Scand. 52 (1983) 200–212zbMATHMathSciNetGoogle Scholar
  49. [KO 1983]
    Koiso, N.: Einstein metrics and complex structure. Invent. math. 73 (1983) 71–106CrossRefzbMATHMathSciNetGoogle Scholar
  50. [KR 1990]
    Kra, I.: Horocyclic coordinates for Riemann surfaces and moduli spaces, I: Teichmüller and Riemann spaces of Kleinian groups. J. Am. Math. Soc. 3 (1990) 499–578zbMATHMathSciNetGoogle Scholar
  51. [LI 1959]
    Lichnerowicz, A.: Isométrie et transformations analytique d’une variété Kählerienne compacte. Bull. Soc. Math. France 87 (1959) 427–437MathSciNetGoogle Scholar
  52. [LI 1978]
    Liebermann, P.: Compactness of the Chow Scheme: application to automorphisms and deformations of Kähler manifolds. Sém Norguet. (Lecture Notes in Mathematics, vol. 670.) Springer, Berlin Heidelberg 1978Google Scholar
  53. [MT 1977]
    Maskit, B.: Decomposition of certain Kleinian groups. Acta math. 130 (1977) 63–82Google Scholar
  54. [M 1957]
    Matsushima, Y.: Sur la structure du groupe d’homomorphismes analytique d’une certaine variété Kählerienne. Nagoya Math. J. 11 (1957) 145–150zbMATHMathSciNetGoogle Scholar
  55. [MA 1976]
    Mazur, H.: The extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43 (1976) 623–635CrossRefMathSciNetGoogle Scholar
  56. [MU 1977]
    Mumford, D.: Stabüity of projective varieties. L’enseign. math. 23 (1977) 39–11zbMATHMathSciNetGoogle Scholar
  57. [PE 1949]
    Petersson, H.: Über die Berechnung der Skalarprodukte ganzer Modulformen. Comment Math. Helv. 22 (1949) 168–199CrossRefzbMATHMathSciNetGoogle Scholar
  58. [PO 1977]
    Popp, H.: Moduli theory and classification theory of algebraic varieties. (Lecture Notes in Mathematics, vol. 620). Springer, Berlin Heidelberg 1977Google Scholar
  59. [RB 1968]
    Richberg, R: Stetig, streng pseudokonvexe Funktionen. Math. Ann. 175 (1968) 257–286CrossRefzbMATHMathSciNetGoogle Scholar
  60. [RE 1985]
    Reich, E.: On the variational principle of Gerstenhaber and Rauch. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985) 469–75CrossRefzbMATHMathSciNetGoogle Scholar
  61. [RI 1857]
    Riemann, B.: Theorie der Abel’schen Functionen. Borchardt’s Journal für reine und angewandte Mathematik, Bd. 54 (1857)Google Scholar
  62. [R01971]
    Royden, H.L.: Automorphisms and isometries of Teichmüller space. Advances in the theory of Riemann surfaces, Stony Brook, 1969. Ann. Math. Studies 66 (1971)Google Scholar
  63. [RO 1974a]
    Royden, H.L.: Invariant metrics on Teichmüller space. In: Ahlfors, L., Kra, I., Maskit, B., Nirenberg, L. (eds.) Contributions to Analysis. New York London, 1974Google Scholar
  64. [RO 1974b]
    Royden, H.L.: Intrinsic metrics on Teichmüller space. Proc. Int. Cong. Math. 2 (1974) 217–221Google Scholar
  65. [SAI 1988]
    Saito, Kyoji: Moduli space for Fuchsian groups. Alg. Analysis II (1988) 735–787Google Scholar
  66. [SA 1978]
    Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Sup. 4 (1978) 211–228MathSciNetGoogle Scholar
  67. [S 1972]
    Schneider, M.: Halbstetigkeitssätze für relativ analytische Räume. Invent. math. 16 (1972) 161–176CrossRefzbMATHMathSciNetGoogle Scholar
  68. [S-Y 1978]
    Schoen, R, Yau, S.T.: On univalent harmonic maps between surfaces. Invent. math 44 (1978) 265–278CrossRefzbMATHMathSciNetGoogle Scholar
  69. [SCH 1983]
    Schumacher, G.: Construction of the coarse moduli space of compact polarized Kähler manifolds with c 1 = 0. Math. Ann. 264 (1983) 81–90CrossRefMathSciNetGoogle Scholar
  70. [SCH 1984]
    Schumacher, G.: Moduli of polarized Kähler manifolds. Math. Ann. 269 (1984) 137–144CrossRefzbMATHMathSciNetGoogle Scholar
  71. [SCH 1986]
    Schumacher, G.: Harmonic maps of the moduli space of compact Riemann surfaces. Math. Ann. 275 (1986) 466–66CrossRefMathSciNetGoogle Scholar
  72. [SCH 1992]
    Schumacher, G.: A remark on the automorphisms of the moduli space M p of compact Riemann surfaces. Arch. Math. 59 (1992) 396–397CrossRefzbMATHMathSciNetGoogle Scholar
  73. [SCH 1993]
    Schumacher, G.: The curvature of the Petersson-Weil metric on the moduli space of Kähler-Einstein manifolds, in Ancona, V. (ed.) et al., Complex analysis and geometry. Plenum Press, New York 1993, pp. 339–354CrossRefGoogle Scholar
  74. [SI 1980]
    Siu, Y.T.: The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. Math. 112 (1980) 73–111CrossRefzbMATHMathSciNetGoogle Scholar
  75. [SI 1986]
    Siu, Y.T.: Curvature of the Weil-Petersson metric in the moduli space of Kähler-Einstein space of negative first Chern class. Aspect of Math. 9. Vieweg, Braunschweig Wiesbaden 1986, pp. 261–298Google Scholar
  76. [SI 1987]
    Siu, Y.T.: Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics. Birkhäuser, Basel Boston 1987CrossRefGoogle Scholar
  77. [TE 1939]
    Teichmüller, O.: Extremale quasikonforme Abbildungen und quadratische Differentiale. Preuß. Akad. math. Wiss., nat. KL 22 (1939) 1–197Google Scholar
  78. [TE 1943]
    Teichmüller, O.: Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen Riemannschen Flächen. Preuß. Akad. math. Wiss., nat. Kl. 4 (1943) 1–42Google Scholar
  79. [TE 1944]
    Teichmüller, O.: Veränderliche Riemannsche Flächen. Deutsche Math. 7 (1944) 344–359zbMATHMathSciNetGoogle Scholar
  80. [TE 1982]
    Teichmüller, O.: Gesammelte Abhandlungen. Collected papers. (Ahlfors, L.V. and Gehring, F.W., eds.). Springer, Berlin Heidelberg 1982Google Scholar
  81. [TI 1987]
    Tian, G.: On Kähler-Einstein metrics on certain Kähler manifolds with C1(M) < 0. Invent, math. 89 (1987) 225–246CrossRefzbMATHMathSciNetGoogle Scholar
  82. [T-Y 1987]
    Tian, G., Yau, S.T.: Kähler-Einstein metrics on complex surfaces with C 1 > 0. Comm. Math. Phys. 112 (1987) 175–203CrossRefzbMATHMathSciNetGoogle Scholar
  83. [TO 1988a]
    Todorov, A.: The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds I. (Preprint)Google Scholar
  84. [TO 1988b]
    Todorov, A.: Weil-Petersson geometry of Teichmüller space of Calabi-Yau manifolds II. (Preprint)Google Scholar
  85. [TR 1986]
    Tromba, A.J.: On a natural affine connection on the space of almost complex structures and the curvature of the Teichmüller space with respect to its Weil-Petersson metric. Man. math. 56 (1986) 475–497CrossRefzbMATHMathSciNetGoogle Scholar
  86. [TR 1987]
    Tromba, A.J.: On an energy function for the Weil-Petersson metric on Teichmüller space. Man. math. 59 (1987) 249–260CrossRefzbMATHMathSciNetGoogle Scholar
  87. [VA 1984]
    Varouchas, J.: Stabilité de la class des variétés Kàhlériennes par certaines morphismes propres. Invent. math. 77 (1984) 117–127CrossRefzbMATHMathSciNetGoogle Scholar
  88. [VA 1989]
    Varouchas, J.: Kàhler spaces and proper open morphisms. Math. Ann. 283 (1989) 13–52CrossRefzbMATHMathSciNetGoogle Scholar
  89. [W 1958]
    Weil, A.: On the moduli of Riemann surfaces. Coll. Works [1958b] Final report on contract AF 18(603)-57; Coll. Works [1958c] Module des surfaces de Riemann, Séminaire Bourbaki, no. 168 (1958)Google Scholar
  90. [W 1989]
    Wolf, M.: The Teichmüller theory of harmonic maps. J. Diff. Geom. 29 (1989) 449–479zbMATHGoogle Scholar
  91. [WO 1983]
    Wolpert, S.: On the homology of the moduli space of stable curves. Ann. Math. 118 (1983) 491–523CrossRefzbMATHMathSciNetGoogle Scholar
  92. [WO 1985a]
    Wolpert, S.: On the Weil-Petersson geometry of the moduli space of curves. Am. J. Math. 107 (1985) 969–997CrossRefzbMATHMathSciNetGoogle Scholar
  93. [WO 1985b]
    Wolpert, S.: On obtaining a positive line bundle from the Weil-Petersson class. Am. J. Math. 107 (1985) 1485–1507CrossRefzbMATHMathSciNetGoogle Scholar
  94. [WO 1986]
    Wolpert, S.: Chern forms and the Riemann Tensor for the moduli space of curves. Invent. math. 85 (1986) 119–145CrossRefzbMATHMathSciNetGoogle Scholar

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  • Georg Schumacher

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