The Fourier transform for certain hyperKähler fourfolds

Abstract

Using a codimension-1 algebraic cycle obtained from the Poincar e line bundle, Beauville de ned the Fourier transform on the Chow groups of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring CH (A). By using a codimension-2 algebraic cycle representing the Beauville-Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperk ahler varieties deformation equivalent to the Hilbert scheme of length-2 subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the Hilbert scheme of length-2 subschemes on a K3 surface and for the variety of lines on a very general cubic fourfold.