Memoirs of the American Mathematical Society 2015; 161 pp; softcover Volume: 240 ISBN10: 1470417405 ISBN13: 9781470417406 List Price: US$90 Individual Members: US$54 Institutional Members: US$72 Order Code: MEMO/240/1139
 Using a codimension\(1\) algebraic cycle obtained from the Poincaré 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 BeauvilleBogomolov class, the authors give evidence for the existence of a similar decomposition for the Chow ring of Hyperkähler varieties deformation equivalent to the Hilbert scheme of length\(2\) 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. Table of Contents The Fourier transform for HyperKähler fourfolds  The Cohomological Fourier Transform
 The Fourier transform on the Chow groups of HyperKähler fourfolds
 The Fourier decomposition is motivic
 First multiplicative results
 An application to symplectic automorphisms
 On the birational invariance of the Fourier decomposition
 An alternate approach to the Fourier decomposition on the Chow ring of Abelian varieties
 Multiplicative ChowKünneth decompositions
 Algebraicity of \(\mathfrak{B}\) for HyperKähler varieties of \(\mathrm{K3}^{[n]}\)type
The Hilbert Scheme \(S^{[2]}\)  Basics on the Hilbert scheme of Length\(2\) subschemes on a variety \(X\)
 The incidence correspondence \(I\)
 Decomposition results on the Chow groups of \(X^{[2]}\)
 Multiplicative ChowKünneth decomposition for \(X^{[2]}\)
 The Fourier decomposition for \(S^{[2]}\)
 The Fourier decomposition for \(S^{[2]}\) is multiplicative
 The Cycle \(L\) of \(S^{[2]}\) via moduli of stable sheaves
The variety of lines on a cubic fourfold  The incidence correspondence \(I\)
 The rational selfmap \(\varphi : F \dashrightarrow F\)
 The Fourier decomposition for \(F\)
 A first multiplicative result
 The rational selfmap \(\varphi :F\dashrightarrow F\) and the Fourier decomposition
 The Fourier decomposition for \(F\) is multiplicative
 Appendix A. Some geometry of cubic fourfolds
 Appendix B. Rational maps and Chow groups
 References
