Demailly’s Holomorphic Morse Inequalities

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


The first aim of this chapter is to provide the background material on differential geometry for the whole book. Then, in the last two sections, we present a heat kernel proof of Demailly’s holomorphic Morse inequalities, Theorem 1.7.1.


Vector Bundle Line Bundle Dirac Operator Heat Kernel Ahler Manifold 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

1.8 Bibliographic notes

  1. [7]
    A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Etudes Sci. Publ. Math. 25 (1965), 81–130; Erratum: 27 (1965), 153–155. p. 113CrossRefMathSciNetGoogle Scholar
  2. [10]
    M.F. Atiyah, R. Bott, and V.K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330, Erratum 28 (1975), 277–280. Appendix IIMATHCrossRefMathSciNetGoogle Scholar
  3. [11]
    M.F. Atiyah and I.M. Singer, Index of elliptic operators. III, Ann. of Math. 87 (1968), 546–604.CrossRefMathSciNetGoogle Scholar
  4. [15]
    N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundl. Math. Wiss. Band 298, Springer-Verlag, Berlin, 1992. § 1.2Google Scholar
  5. [24]
    J.-M. Bismut, The Witten complex and degenerate Morse inequalities, J. Differential Geom. 23 (1986), 207–240.MATHMathSciNetGoogle Scholar
  6. [25]
    _____, Demailly’s asymptotic inequalities: a heat equation proof, J. Funct. Anal. 72 (1987), 263–278.MATHCrossRefMathSciNetGoogle Scholar
  7. [26]
    _____, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), no. 4, 681–699.MATHCrossRefMathSciNetGoogle Scholar
  8. [33]
    J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. (1991), no. 74, ii+298 pp. (1992). §11Google Scholar
  9. [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. Th. 1.1MATHCrossRefMathSciNetGoogle Scholar
  10. [43]
    D. Borthwick and A. Uribe, Almost complex structures and geometric quantization, Math. Res. Lett. 3 (1996), 845–861. Erratum: Math. Res. Lett. 5 (1998), 211–212.MATHMathSciNetGoogle Scholar
  11. [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
  12. [54]
    M. Braverman, Vanishing theorems on covering manifolds, Contemp. Math. 213 (1999), 1–23.Google Scholar
  13. [72]
    J.-P. Demailly, Champs magnétiques et inegalités de Morse pour la d″-cohomologie, Ann. Inst. Fourier (Grenoble) 35 (1985), 189–229.MATHMathSciNetGoogle Scholar
  14. [73]
    _____, Sur l’identité de Bochner-Kodaira-Nakano en géométrie hermitienne, Lecture Notes in Math., vol. 1198, pp. 88–97, Springer Verlag, 1985.MathSciNetGoogle Scholar
  15. [74]
    _____, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 93–114.Google Scholar
  16. [85]
    M. do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992.Google Scholar
  17. [108]
    G.B. Folland and J.J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton, N.J., 1972, Annals of Mathematics Studies, No. 75.MATHGoogle Scholar
  18. [119]
    P. Griffiths, The extension problem in complex analysis; embedding with positive normal bundle, Amer. J. Math. 88 (1966), 366–446.MATHCrossRefMathSciNetGoogle Scholar
  19. [130]
    F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9, Springer-Verlag, Berlin, 1956.Google Scholar
  20. [131]
    L. Hörmander, L 2-estimates and existence theorem for the \( \bar \partial \)-operator, Acta Math. 113 (1965), 89–152.MATHCrossRefMathSciNetGoogle Scholar
  21. [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
  22. [143]
    J.J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds I, Ann. Math. 78 (1963), 112–148.CrossRefMathSciNetGoogle Scholar
  23. [148]
    H.B. Lawson and M.-L. Michelson, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton Univ. Press, Princeton, NJ, 1989. Appendix IIGoogle Scholar
  24. [160]
    X. Ma and G. Marinescu, The Spinc Dirac operator on high tensor powers of a line bundle, Math. Z. 240 (2002), no. 3, 651–664. § 2MATHCrossRefMathSciNetGoogle Scholar
  25. [175]
    V. Mathai and S. Wu, Equivariant holomorphic Morse inequalities. I. Heat kernel proof, J. Differential Geom. 46 (1997), no. 1, 78–98.MATHMathSciNetGoogle Scholar
  26. [176]
    J. Milnor, Morse Theory, Ann. Math. Studies, vol. 51, Princeton University Press, 1963.Google Scholar
  27. [179]
    J. Morrow and K. Kodaira, Complex manifolds, AMS Chelsea Publishing, Providence, RI, 2006, Reprint of the 1971 edition with errata.MATHGoogle Scholar
  28. [187]
    T. Ohsawa, Isomorphism theorems for cohomology groups of weakly 1-complete manifolds, Publ. Res. Inst. Math. Sci. 18 (1982), 191–232.MATHMathSciNetGoogle Scholar
  29. [251]
    A. Weil, Introduction a l’étude des variétés kählériennes, Actualités scientifiques et industrielles, vol. 1267, Hermann, Paris, 1958.Google Scholar
  30. [252]
    R.O. Wells, Differential analysis on complex manifolds, second ed., GTM, vol. 65, Springer-Verlag, New York, 1980.MATHGoogle Scholar
  31. [253]
    E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661–692.MATHMathSciNetGoogle Scholar
  32. [254]
    _____, Holomorphic Morse inequalities, Algebraic and differential topology — global differential geometry, Teubner-Texte Math., vol. 70, Teubner, Leipzig, 1984, pp. 318–333.Google Scholar
  33. [256]
    S. Wu and W. Zhang, Equivariant holomorphic Morse inequalities. III. Non-isolated fixed points, Geom. Funct. Anal. 8 (1998), no. 1, 149–178.MATHCrossRefMathSciNetGoogle Scholar
  34. [263]
    W. Zhang, Lectures on Chern-Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, vol. 4, World Scientific, Hong Kong, 2001.Google Scholar

Copyright information

© Birkhäuser Verlag AG 2007

Personalised recommendations