Vector bundles on Abelian Varieties

Abstract

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 Poincaré 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 Poincaré bundle on X × $\hat X$, and p1 and p2 are the projections of X × $\hat X$, then

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

is a cycle class on $\hat X$. Similarly, if ε is a coherent sheaf on X, then

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

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 ε 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

$$DS\left( \varepsilon \right) = Rp{2_*}\left( {P \otimes p_1^*\varepsilon } \right)$$

is an element of the analogous derived category D ^b ($\hat X$) of the dual abelian variety. Mukai showed that this functor is an equivalence of categories and gave several applications. He called it the Fourierfunctor and DS(·) the Fourier transform because of its formal analogy to the Fourier transform in Analysis.