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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
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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
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