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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
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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

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.

• 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