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The Fourier Transform for Certain HyperKähler Fourfolds
Mingmin Shen, Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands, and Charles Vial, University of Cambridge, United Kingdom
 Memoirs of the American Mathematical Society 2015; 161 pp; softcover Volume: 240 ISBN-10: 1-4704-1740-5 ISBN-13: 978-1-4704-1740-6 List Price: US$90 Individual Members: US$54 Institutional Members: US\$72 Order Code: MEMO/240/1139 Not yet published.Expected publication date is March 2, 2016. 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 Beauville-Bogomolov 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 Introduction 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 Chow-Künneth 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 Chow-Künneth 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 self-map $$\varphi : F \dashrightarrow F$$ The Fourier decomposition for $$F$$ A first multiplicative result The rational self-map $$\varphi :F\dashrightarrow F$$ 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