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A Note on Cohomology Groups of Holomorphic Line Bundles over a Complex Torus

  • Shingo Murakami
Part of the Progress in Mathematics book series (PM, volume 14)

Abstract

Let E = V/L be an n-dimensional complex torus, where V is an n-dimensional complex vector space and L a lattice of V. Let F be a holomorphic line bundle over E. In this note, we shall show that the q-th cohomology group Hq(E,F) (q ≧ 0) of E with coefficients in the sheaf F of germs of holomorphic sections of F can be completely determined by applying harmonic theory. The results have been obtained by Mumford [3] and Kempf [1] by an algebraico-geometric way, and later Umemura [4] and Matsushima [2] showed that harmonic theory can be used to get the results provided that the line bundle F has nondegenerate Chern class. Our purpose is to note that the method found by Umemura and Matsushima may be modified so as to prove a structure theorem in the general case.

Keywords

Line Bundle Cohomology Group Theta Function Hermitian Form Holomorphic Section 
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.

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References

  1. [1]
    G. Kempf, Appendix to D. Mumford’s article: “Varieties defined by quadratic equations,” Questions in Algebraic Varieties, C.I.M.E., 1969.Google Scholar
  2. [2]
    Y. Matsushima, “On the intermediate cohomology group of a holomorphic line bundle over a complex torus,” Osaka J. Math, 16 (1979), 617–632.MathSciNetMATHGoogle Scholar
  3. [3]
    D. Mumford, Abelian Varieties, Tata Institute Studies in Math., Oxford Univ. Press, 1970.MATHGoogle Scholar
  4. [4]
    H. Umemura, “Some results in the theory of vector bundles,” Nagoya Math. J. 52 (1973), 97–128.MathSciNetMATHGoogle Scholar
  5. [5]
    A. Weil, Variétés Kähleriennes, Hermann, Paris, 1958.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Shingo Murakami
    • 1
  1. 1.Osaka UniversityToyonaka, Osaka 560Japan

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