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.

Supported by Grant-in-Aid for Scientific Research.