Christina Birkenhake

Vector bundles on Abelian Varieties

Vector bundles on abelian varieties have been studied almost as long as on any other variety. In one of the first papers on the subject Atiyah [1] classified vector bundles on elliptic curves. There were some scattered results on homogeneous vector bundles. The fundamental idea on the subject however is due to Mukai [1]. In the theory of algebraic cycles it is quite common to use the Poincare bundle to transfer cycles on an abelian variety X to cycles on the dual abelian variety (see e.g. Weil [1]). To be more precise: If a is a cycle class on X, P denotes the Poincare bundle on X × \(\hat X\), and p1 and p2 are the projections of X × \(\hat X\), then

Cycles on Abelian varieties

S\left( a \right): = p{2_*}\left( {c1\left[ P \right]} \right)\cdot p_1^*a

Equations for Abelian Varieties

is a cycle class on \(\hat X\). Similarly, if e is a coherent sheaf on X, then

An isomorphism between moduli spaces of abelian varieties

S\left( \varepsilon \right): = p{2_*}\left( {P \otimes p_1^*\varepsilon } \right)

Nonplussed! mathematical proof of implausible ideas

is a coherent sheaf on \(\hat X\). In general this sheaf is not very useful. Is was Mukai who saw that a modification of this sheaf is of considerable importance. Namely, consider e as an element, or more generally consider any element, of the derived category D b (X) of bounded complexes of the category of coherent O X —modules, then