The Fourier Transform for Certain HyperKähler Fourfolds

Abstract

Using a codimension- 1 1 algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety A A and showed that the Fourier transform induces a decomposition of the Chow ring C H ∗ ( A ) \mathrm {CH}^*(A) . By using a codimension- 2 2 algebraic cycle representing the Beauville–Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperKähler varieties deformation equivalent to the Hilbert scheme of length- 2 2 subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the Hilbert scheme of length- 2 2 subschemes on a K3 surface and for the variety of lines on a very general cubic fourfold.