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.
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Murre, J.P. (2000). Algebraic Cycles on Abelian Varieties. In: Gordon, B.B., Lewis, J.D., Müller-Stach, S., Saito, S., Yui, N. (eds) The Arithmetic and Geometry of Algebraic Cycles. NATO Science Series, vol 548. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4098-0_11
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DOI: https://doi.org/10.1007/978-94-011-4098-0_11
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