A Note on Cohomology Groups of Holomorphic Line Bundles over a Complex Torus
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  and Kempf  by an algebraico-geometric way, and later Umemura  and Matsushima  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.
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- G. Kempf, Appendix to D. Mumford’s article: “Varieties defined by quadratic equations,” Questions in Algebraic Varieties, C.I.M.E., 1969.Google Scholar