Advertisement

Kodaira Map

Part of the Progress in Mathematics book series (PM, volume 254)

Abstract

In this chapter we present some applications of the asymptotic expansion of the Bergman kernel.

Keywords

Vector Bundle Line Bundle Einstein Metrics Bergman Kernel Holomorphic Line Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

5.6 Bibliographic notes

  1. [12]
    W. Baily, The decomposition theorem for V-manifolds, Amer. J. Math. 78 (1956), 862–888.MATHCrossRefMathSciNetGoogle Scholar
  2. [13]
    W. Baily, On the imbedding of V-manifolds in projective space, Amer. J. Math. 79 (1957), 403–430.MATHCrossRefMathSciNetGoogle Scholar
  3. [14]
    W. Baily, Satake’s compactification of Vn, Amer. J. Math. 80 (1958), 348–364.MATHCrossRefMathSciNetGoogle Scholar
  4. [21]
    B. Berndtsson, Positivity of direct image bundles and convexity on the space of Kähler metrics, (2006), math.CV/0608385.Google Scholar
  5. [23]
    O. Biquard, Métriques kählériennes à courbure scalaire constante, Astérisque (2006), no. 307, Exp. No. 938, Séminaire Bourbaki. Vol. 2004/2005.Google Scholar
  6. [28]
    J.-M. Bismut, From Quillen metrics to Reidemeister metrics: some aspects of the Ray-Singer analytic torsion, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 273–324.Google Scholar
  7. [30]
    J.-M. Bismut, Local index theory and higher analytic torsion, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, pp. 143–162 (electronic).MathSciNetGoogle Scholar
  8. [31]
    J.-M. Bismut, Local index theory, eta invariants and holomorphic torsion: a survey, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), Int. Press, Boston, MA, 1998, pp. 1–76.Google Scholar
  9. [32]
    J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Comm. Math. Phys. 115 (1988), no. 2, 301–351. Theorem 1.18CrossRefMathSciNetGoogle Scholar
  10. [34]
    J.-M. Bismut and X. Ma, Holomorphic immersions and equivariant torsion forms, J. Reine Angew. Math. 575 (2004), 189–235.MATHMathSciNetGoogle Scholar
  11. [35]
    J.-M. Bismut and E. Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), 355–367.MATHCrossRefMathSciNetGoogle Scholar
  12. [36]
    J.-M. Bismut and E. Vasserot, The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 835–848.MATHMathSciNetGoogle Scholar
  13. [37]
    P. Bleher, B. Shiffman, and S. Zelditch, Poincaré-Lelong approach to universality and scaling of correlations between zeros, Comm. Math. Phys. 208 (2000), no. 3, 771–785.MATHCrossRefMathSciNetGoogle Scholar
  14. [38]
    P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351–395.MATHCrossRefMathSciNetGoogle Scholar
  15. [48]
    Th. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéare positif, Ann. Inst. Fourier (Grenoble) 40 (1990), 117–130.MATHMathSciNetGoogle Scholar
  16. [49]
    Th. Bouche, Asymptotic results for Hermitian line bundles over complex manifolds: the heat kernel approach, Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 67–81.Google Scholar
  17. [60]
    H. Cartan, Séminaires de H. Cartan, 1953–1954.Google Scholar
  18. [61]
    H. Cartan, Quotient d’un espace analytique par un groupe d’automorphismes, Algebraic geometry and topology., Princeton University Press, Princeton, N. J., 1957, A symposium in honor of S. Lefschetz,, pp. 90–102.Google Scholar
  19. [62]
    D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, MA, 1999, pp. 1–23.Google Scholar
  20. [69]
    X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1–41; announced in C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198.MATHMathSciNetGoogle Scholar
  21. [79]
    J.-P. Demailly, Complex analytic and differential geometry, 2001, published online at www-fourier.ujf-grenoble.fr/?demailly/lectures.html. II.3.22Google Scholar
  22. [84]
    T.-C. Dinh and N. Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (2006), no. 1, 221–258.MATHMathSciNetCrossRefGoogle Scholar
  23. [86]
    S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), no. 1, 231–247.MATHCrossRefMathSciNetGoogle Scholar
  24. [90]
    S.K. Donaldson, Scalar curvature and projective embeddings, J. Differential Geom. 59 (2001), no. 2, 479–522.MATHMathSciNetGoogle Scholar
  25. [91]
    S.K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349.MATHMathSciNetGoogle Scholar
  26. [92]
    S.K. Donaldson, Conjectures in Kähler geometry, Strings and geometry, Clay Math. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 2004, pp. 71–78.Google Scholar
  27. [93]
    S.K. Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), no. 3, 453–472.MATHMathSciNetGoogle Scholar
  28. [94]
    S.K. Donaldson, Scalar curvature and projective embeddings. II, Q. J. Math. 56 (2005), no. 3, 345–356.CrossRefMathSciNetMATHGoogle Scholar
  29. [95]
    S.K. Donaldson, Some numerical results in complex differential geometry, (2006), math.DG/0512625.Google Scholar
  30. [97]
    M. Douglas, B. Shiffman, and S. Zelditch, Critical points and supersymmetric vacua. I, Comm. Math. Phys. 252 (2004), no. 1–3, 325–358.MATHCrossRefMathSciNetGoogle Scholar
  31. [98]
    M. Douglas, B. Shiffman, and S. Zelditch, Critical points and supersymmetric vacua. II. Asymptotics and extremal metrics, J. Differential Geom. 72 (2006), no. 3, 381–427.MATHMathSciNetGoogle Scholar
  32. [99]
    M. Douglas, B. Shiffman, and S. Zelditch, Critical points and supersymmetric vacua. III. String/M models, Comm. Math. Phys. 265 (2006), no. 3, 617–671.MATHCrossRefMathSciNetGoogle Scholar
  33. [107]
    G. Fischer, Complex analytic geometry, Lect. Notes, vol. 538, Springer-Verlag, 1976. p. 76Google Scholar
  34. [111]
    P. Gauduchon, Calabi’s extremal Kähler metrics: an elementary introduction, 2005, book in preparation.Google Scholar
  35. [124]
    R. Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin, 1977. III.4.5MATHGoogle Scholar
  36. [132]
    L. Hörmande, An introduction to complex analysis in several variables, 1966, third ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
  37. [136]
    T. Kawasaki, The Riemann-Roch theorem for complex V-manifolds, Osaka J. Math. 16 (1979), no. 1, 151–159.MATHMathSciNetGoogle Scholar
  38. [137]
    T. Kawasaki, The index of elliptic operators over V-manifolds, Nagoya Math. J. 84 (1981), 135–157.MATHMathSciNetGoogle Scholar
  39. [138]
    J. Keller, Vortex type equations and canonical metrics, math.DG/0601485, 2006.Google Scholar
  40. [140]
    S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ, 1987, Kanô Memorial Lectures, 5.MATHGoogle Scholar
  41. [141]
    K. Kodaira, On Kähler varieties of restricted type, Ann. of Math. 60 (1954), 28–48.CrossRefMathSciNetGoogle Scholar
  42. [142]
    K. Köhler and D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof, Invent. Math. 145 (2001), no. 2, 333–396.MATHCrossRefMathSciNetGoogle Scholar
  43. [154]
    N. Leung, Einstein type metrics and stability on vector bundles, J. Differential Geom. 45 (1997), no. 3, 514–546.MATHMathSciNetGoogle Scholar
  44. [158]
    H. Luo, Geometric criterion for Gieseker-Mumford stability of polarized manifolds, J. Differential Geom. 49 (1998), no. 3, 577–599.MATHMathSciNetGoogle Scholar
  45. [159]
    X. Ma, Orbifolds and analytic torsions, Trans. Amer. Math. Soc. 357 (2005), no. 6, 2205–2233.MATHCrossRefMathSciNetGoogle Scholar
  46. [165]
    X. Ma and W. Zhang, Superconnection and family Bergman kernels, C. R. Math. Acad. Sci. Paris 344 (2007), 41–44.MATHMathSciNetGoogle Scholar
  47. [166]
    T. Mabuchi, An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I, Invent. Math. 159 (2005), no. 2, 225–243.MATHCrossRefMathSciNetGoogle Scholar
  48. [167]
    T. Mabuchi, The Chow-stability and Hilbert-stability in Mumford’s geometric invariant theory, (2006), math.DG/0607590.Google Scholar
  49. [168]
    T. Mabuchi, An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. II, J. Differential Geom. (2006), to appear.Google Scholar
  50. [185]
    Y. Nohara, Projective embeddings and Lagrangian fibrations of Kummer varieties, (2006), math.DG/0604329.Google Scholar
  51. [192]
    S.T. Paul and G. Tian, Algebraic and analytic K-stability, math.DG/0405530.Google Scholar
  52. [193]
    S.T. Paul and G. Tian, CM stability and the generalised Futaki invariant, math.AG/0605278.Google Scholar
  53. [194]
    S.T. Paul and G. Tian, Analysis of geometric stability, Int. Math. Res. Not. (2004), no. 48, 2555–2591.CrossRefMathSciNetGoogle Scholar
  54. [197]
    D. Phong and J. Sturm, Scalar curvature, moment maps, and the Deligne pairing, Amer. J. Math. 126 (2004), no. 3, 693–712.MATHCrossRefMathSciNetGoogle Scholar
  55. [198]
    D. Phong and J. Sturm, The Monge-Ampère operator and geodesics in the space of Kähler potentials, Invent. Math. 166 (2006), no. 1, 125–149.MATHCrossRefMathSciNetGoogle Scholar
  56. [199]
    D. Phong and J. Sturm, Test configurations for K-stability and geodesic rays, math.DG/0606423 (2006).Google Scholar
  57. [200]
    D. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Functional Anal. Appl. 19 (1985), no. 1, 31–34.MATHCrossRefMathSciNetGoogle Scholar
  58. [201]
    D.B. Ray and I.M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177.CrossRefMathSciNetGoogle Scholar
  59. [205]
    J. Ross and R. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differential Geom. 72 (2006), no. 3, 429–466.MATHMathSciNetGoogle Scholar
  60. [206]
    J. Ross and R. Thomas, A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), no. 2, 201–255.MATHMathSciNetGoogle Scholar
  61. [208]
    W.-D. Ruan, Canonical coordinates and Bergmann metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589–631.MATHMathSciNetGoogle Scholar
  62. [210]
    I. Satake, The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan 9 (1957), 464–492.MATHMathSciNetCrossRefGoogle Scholar
  63. [214]
    J.P. Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955), 197–278.CrossRefMathSciNetGoogle Scholar
  64. [215]
    J.P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956), 1–42.MathSciNetGoogle Scholar
  65. [218]
    B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (1999), no. 3, 661–683.MATHCrossRefMathSciNetGoogle Scholar
  66. [220]
    B. Shiffman and S. Zelditch, Equilibrium distribution of zeros of random polynomials, Int.Math. Res. Not. (2003), no. 1, 25–49.CrossRefMathSciNetGoogle Scholar
  67. [221]
    B. Shiffman and S. Zelditch, Random polynomials with prescribed Newton polytope, J. Amer. Math. Soc. 17 (2004), no. 1, 49–108.MATHCrossRefMathSciNetGoogle Scholar
  68. [227]
    Y.T. Siu and S.T. Yau, Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay, Ann. of Math. (2) 105 (1977), no. 2, 225–264.CrossRefMathSciNetGoogle Scholar
  69. [230]
    J. Song, The Szegö kernel on an orbifold circle bundle, (2004), math.DG/0405071.Google Scholar
  70. [231]
    J. Song, The α-invariant on toric Fano manifolds, Amer. J. Math. 127 (2005), no. 6, 1247–1259.MATHCrossRefMathSciNetGoogle Scholar
  71. [232]
    C. Soulé, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992.Google Scholar
  72. [233]
    C. Soulé, Genres de Todd et valeurs aux entiers des dérivées de fonctions L, Astérisque (2007), no. 311, Exp. No. 955,, Séminaire Bourbaki, Vol. 2005/2006.Google Scholar
  73. [240]
    R. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, Survey in Differential Geometry, X (2006), 221–273.Google Scholar
  74. [241]
    G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99–130. Theorem AMATHMathSciNetGoogle Scholar
  75. [242]
    G. Tian, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000, Notes taken by Meike Akveld.MATHGoogle Scholar
  76. [243]
    G. Tian, Extremal metrics and geometric stability, Houston J. Math. 28 (2002), no. 2, 411–432, Special issue for S. S. Chern.MATHMathSciNetGoogle Scholar
  77. [246]
    K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257–S293.CrossRefMathSciNetGoogle Scholar
  78. [247]
    L. Wang, Bergman kernel and stability of holomorphic vector bundles with sections, MIT Ph.D. Thesis (2003), 85 pages.Google Scholar
  79. [248]
    X. Wang, Balance point and stability of vector bundles over a projective manifold, Math. Res. Lett. 9 (2002), no. 2–3, 393–411.MATHMathSciNetGoogle Scholar
  80. [249]
    X. Wang, Moment map, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004), no. 5, 1009–1037.MATHMathSciNetGoogle Scholar
  81. [250]
    X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005), no. 2, 253–285.MATHMathSciNetGoogle Scholar
  82. [258]
    S.-T. Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1–2, 109–158.MATHMathSciNetGoogle Scholar
  83. [259]
    S.-T. Yau, Perspectives on geometric analysis, Survey in Differential Geometry, X (2006), 275–379.Google Scholar
  84. [260]
    K.-I. Yoshikawa, K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space, Invent. Math. 156 (2004), no. 1, 53–117.MATHCrossRefMathSciNetGoogle Scholar
  85. [261]
    S. Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), no. 6, 317–331.CrossRefMathSciNetGoogle Scholar
  86. [262]
    S. Zhang, Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), no. 1, 77–105.MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag AG 2007

Personalised recommendations