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)


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.


Manifold Zine 


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  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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