Abstract
Publisher Summary This chapter reviews the results on algebraic vector bundles. In the theory of algebraic vector bundles, one of the most important and difficult problems is to find a fine procedure to construct vector bundles. There are several methods to construct vector bundles. This fact shows that every vector bundle on a nonsingular quasi-projective variety is a successive extension of line bundles after blowing-ups with smooth centers of the base variety. The construction of vector bundles in the theorem is by means of elementary transformation stated in the preceding section. In this study of vector bundles on abelian varieties, especially an elliptic curve, first looked into the vector bundles constructed by taking direct image of line bundles by isogenies. This method is superior to Atiyah's [A], successive extensions of line bundles, at least, to handle the pull-back by the Frobenius map F.
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