ある種の楕円曲面上の複素平面束について

その他

M. F. Atiyah has given the cuassification theorem for holomorphic vector bundles over an elliptic curve, (Theorem 7, [2]). In the proof, two lemmas are effective, which are called the uniqueness and existence theorems. These are the motive for this paper. In §1, we prove that, over a product surface of a non singular curve and an elliptic curve, if a line bundle satisfies some condltion about a local triviality and the Chern class, then it admits a non trivial extension to a-plane bundle. This fact corresponds to Lemma 16, [2] . In §2, we define a strongly reducible plane bundle and prove that not every plane bundle is strongly reducible over a basic member (8, [4]) on an algebraic curve of genus greater than one. This fact corresponds to Lemma 15, [2].