Download The Fourier Transform for Certain Hyperkahler Fourfolds (Memoirs of the American Mathematical Society)
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Using a codimension-$1$ algebraic cycle obtained from the Poincare line bundle, Beauville defined 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 $\mathrm{CH}^*(A)$. By using a codimension-$2$ algebraic cycle representing the Beauville-Bogomolov class, the authors give evidence for the existence of a similar decomposition for the Chow ring of Hyperkahler varieties deformation equivalent to the Hilbert scheme of length-$2$ subschemes on a K3 surface. They 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.
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