Algebraic Cycles on Abelian Varieties

Abstract

We give a survey of the Fourier-Mukai transform on abelian varieties. This is a correspondence from an abelian variety to its dual abelian variety, constructed from the Poincaré bundle. This correspondence was used by Lieberman and Mukai to compute cohomology and K -theory of abelian varieties and later by Beauville to study Chow groups of abelian varieties. We discuss the main theorem and the essential part of its proof (the so-called inversion formula) and as applications a theorem of Bloch on Pontryagin powers of algebraic cycles and the decomposition theorem of Beauville for Chow groups. We conclude by mentioning some further developments due to Deninger-Murre and Künnemann.KeywordsAbelian VarietyDecomposition TheoremChern CharacterDivisor ClassChow GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.