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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2024 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Book Review

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MathSciNet review: 2729895
Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Xiaonan Ma and George Marinescu
Title: Holomorphic Morse inequalities and Bergman kernels
Additional book information: Progress in Mathematics, 254, Birkhäuser Verlag, Basel, 2007, xiv+422 pp., ISBN 978-3-7643-8096-0, US$79.95$

References [Enhancements On Off] (What's this?)

  • Robert Berman, Bergman kernels and local holomorphic Morse inequalities, Math. Z. 248 (2004), no. 2, 325–344. MR 2088931, DOI 10.1007/s00209-003-0630-z
    • [B2]
    R. Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math. (to appear); arXiv:0710.4375.
    [BBSj]
    R. Berman, B. Berndtsson and J. Sjöstrand, Asymptotics of Bergman kernels, Amer. J. Math. (to appear); arXiv: math/0506367v2.
  • Jean-Michel Bismut, Demailly’s asymptotic Morse inequalities: a heat equation proof, J. Funct. Anal. 72 (1987), no. 2, 263–278. MR 886814, DOI 10.1016/0022-1236(87)90089-9
  • Pavel Bleher, Bernard Shiffman, and Steve Zelditch, Universality and scaling of zeros on symplectic manifolds, Random matrix models and their applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 31–69. MR 1842782
  • Pavel Bleher, Bernard Shiffman, and Steve Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351–395. MR 1794066, DOI 10.1007/s002220000092
  • David Borthwick and Alejandro Uribe, Nearly Kählerian embeddings of symplectic manifolds, Asian J. Math. 4 (2000), no. 3, 599–620. MR 1796696, DOI 10.4310/AJM.2000.v4.n3.a6
  • L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegő, Journées: Équations aux Dérivées Partielles de Rennes (1975), Astérisque, No. 34-35, Soc. Math. France, Paris, 1976, pp. 123–164 (French). MR 0590106
  • L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR 620794, DOI 10.1515/9781400881444
  • David 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. MR 1699887
  • Jean-Pierre Demailly, 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. MR 1128538, DOI 10.4310/pamq.2011.v7.n4.a6
    • [Dem2]
    J. P. Demailly, Complex analytic and algebraic geometry. Available at http://www-fourier.ujf-grenoble.fr/ demailly/books.html.
  • S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666–705. MR 1438190
  • S. K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), no. 3, 479–522. MR 1916953
  • Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
  • Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
  • V. Guillemin and A. Uribe, The Laplace operator on the $n$th tensor power of a line bundle: eigenvalues which are uniformly bounded in $n$, Asymptotic Anal. 1 (1988), no. 2, 105–113. MR 950009
  • Shanyu Ji and Bernard Shiffman, Properties of compact complex manifolds carrying closed positive currents, J. Geom. Anal. 3 (1993), no. 1, 37–61. MR 1197016, DOI 10.1007/BF02921329
  • Niklas Lindholm, Sampling in weighted $L^p$ spaces of entire functions in ${\Bbb C}^n$ and estimates of the Bergman kernel, J. Funct. Anal. 182 (2001), no. 2, 390–426. MR 1828799, DOI 10.1006/jfan.2000.3733
  • Zhiqin Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235–273. MR 1749048
  • A. Menikoff and J. Sjöstrand, On the eigenvalues of a class of hypoelliptic operators, Math. Ann. 235 (1978), no. 1, 55–85. MR 481627, DOI 10.1007/BF01421593
  • K. Wendland and W. Müller, Critical metrics with respect to Ray-Singer analytic torsion and Quillen metric, Analysis, numerics and applications of differential and integral equations (Stuttgart, 1996) Pitman Res. Notes Math. Ser., vol. 379, Longman, Harlow, 1998, pp. 245–250. MR 1606714
  • Duong H. Phong and Jacob Sturm, Test configurations for K-stability and geodesic rays, J. Symplectic Geom. 5 (2007), no. 2, 221–247. MR 2377252
  • D. H. Phong and Jacob Sturm, The Monge-Ampère operator and geodesics in the space of Kähler potentials, Invent. Math. 166 (2006), no. 1, 125–149. MR 2242635, DOI 10.1007/s00222-006-0512-1
  • Bernard Shiffman and Andrew John Sommese, Vanishing theorems on complex manifolds, Progress in Mathematics, vol. 56, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 782484, DOI 10.1007/978-1-4899-6680-3
  • Bernard Shiffman and Steve Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181–222. MR 1887895, DOI 10.1515/crll.2002.023
  • Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. MR 1064867
  • Xiaowei Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005), no. 2, 253–285. MR 2154820
  • Steve Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317–331. MR 1616718, DOI 10.1155/S107379289800021X
    • [Z2]
    S. Zelditch. Bernstein polynomials, Bergman kernels, and toric Kähler varieties, J. Symplectic Geom. (to appear); arXiv: 0705.2879.
    Review Information:

    Reviewer: Steve Zelditch
    Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, 21218
    Email: zelditch@math.jhu.edu
    Journal: Bull. Amer. Math. Soc. 46 (2009), 349-361
    DOI: https://doi.org/10.1090/S0273-0979-08-01224-X
    Published electronically: October 14, 2008
    Additional Notes: Research partially supported by NSF grant DMS-0603850.
    Review copyright: © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.