Manifolds and Lie Groups pp 301-313 | Cite as

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

## 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 H^{q}(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## Preview

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## References

- [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]Y. Matsushima, “On the intermediate cohomology group of a holomorphic line bundle over a complex torus,”
*Osaka J. Math*, 16 (1979), 617–632.MathSciNetzbMATHGoogle Scholar - [3]D. Mumford,
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*Nagoya Math. J.*52 (1973), 97–128.MathSciNetzbMATHGoogle Scholar - [5]A. Weil,
*Variétés Kähleriennes*, Hermann, Paris, 1958.zbMATHGoogle Scholar