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The Fourier Transform for Certain HyperKähler Fourfolds

About this Title

Mingmin Shen and Charles Vial

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 240, Number 1139
ISBNs: 978-1-4704-1740-6 (print); 978-1-4704-2830-3 (online)
Published electronically: November 18, 2015
Keywords:HyperKähler manifolds, irreducible holomorphic symplectic varieties, cubic fourfolds, Fano schemes of lines, K3 surfaces, Hilbert schemes of points, Abelian varieties, motives, algebraic cycles, Chow groups, Chow ring, Chow–Künneth decomposition, Bloch–Beilinson filtration, modified diagonals

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Table of Contents


  • Introduction
  • Chapter 1. The Cohomological Fourier Transform
  • Chapter 2. The Fourier Transform on the Chow Groups of HyperKähler Fourfolds
  • Chapter 3. The Fourier Decomposition Is Motivic
  • Chapter 4. First Multiplicative Results
  • Chapter 5. An Application to Symplectic Automorphisms
  • Chapter 6. On the Birational Invariance of the Fourier Decomposition
  • Chapter 7. An Alternate Approach to the Fourier Decomposition on the Chow Ring of Abelian Varieties
  • Chapter 8. Multiplicative Chow–Künneth Decompositions
  • Chapter 9. Algebraicity of $\mathfrak B$ for HyperKähler Varieties of $\mathrm {K3}^{[n]}$-type
  • Chapter 10. Basics on the Hilbert Scheme of Length-$2$ Subschemes on a Variety $X$
  • Chapter 11. The Incidence Correspondence $I$
  • Chapter 12. Decomposition Results on the Chow Groups of $X^{[2]}$
  • Chapter 13.
  • Chapter 14. The Fourier Decomposition for $S^{[2]}$
  • Chapter 15. The Fourier Decomposition for $S^{[2]}$ is Multiplicative
  • Chapter 16. The Cycle $L$ of $S^{[2]}$ via Moduli of Stable Sheaves
  • Chapter 17. The Incidence Correspondence $I$
  • Chapter 18. The Rational Self-Map $\varphi : F \dashrightarrow F$
  • Chapter 19. The Fourier Decomposition for $F$
  • Chapter 20. A First Multiplicative Result
  • Chapter 21. The Rational Self-Map $\varphi :F\dashrightarrow F$ and the Fourier Decomposition
  • Chapter 22. The Fourier Decomposition for $F$ is Multiplicative
  • Appendix A. Some Geometry of Cubic Fourfolds
  • Appendix B. Rational Maps and Chow Groups


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 $CH^{*}(A)$. By using a codimension-$2$ algebraic cycle representing the Beauville–Bogomolov class, we 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. We 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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