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The Fourier Transform for Certain HyperKähler Fourfolds
 
Mingmin Shen Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
Charles Vial University of Cambridge, United Kingdom
The Fourier Transform for Certain HyperK\"ahler Fourfolds
eBook ISBN:  978-1-4704-2830-3
Product Code:  MEMO/240/1139.E
List Price: $90.00
MAA Member Price: $81.00
AMS Member Price: $54.00
The Fourier Transform for Certain HyperK\"ahler Fourfolds
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The Fourier Transform for Certain HyperKähler Fourfolds
Mingmin Shen Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
Charles Vial University of Cambridge, United Kingdom
eBook ISBN:  978-1-4704-2830-3
Product Code:  MEMO/240/1139.E
List Price: $90.00
MAA Member Price: $81.00
AMS Member Price: $54.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2402015; 161 pp
    MSC: Primary 14; 53;

    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
     
     
    • Chapters
    • Introduction
    • 1. The Fourier Transform for HyperKähler Fourfolds
    • 1. The Cohomological Fourier Transform
    • 2. The Fourier Transform on the Chow Groups of HyperKähler Fourfolds
    • 3. The Fourier Decomposition Is Motivic
    • 4. First Multiplicative Results
    • 5. An Application to Symplectic Automorphisms
    • 6. On the Birational Invariance of the Fourier Decomposition
    • 7. An Alternate Approach to the Fourier Decomposition on the Chow Ring of Abelian Varieties
    • 8. Multiplicative Chow–Künneth Decompositions
    • 9. Algebraicity of $\mathfrak {B}$ for HyperKähler Varieties of $\mathrm {K3}^{[n]}$-type
    • 2. The Hilbert Scheme $S^{[2]}$
    • 10. Basics on the Hilbert Scheme of Length-$2$ Subschemes on a Variety $X$
    • 11. The Incidence Correspondence $I$
    • 12. Decomposition Results on the Chow Groups of $X^{[2]}$
    • 13. Multiplicative Chow–Künneth Decomposition for $X^{[2]}$
    • 14. The Fourier Decomposition for $S^{[2]}$
    • 15. The Fourier Decomposition for $S^{[2]}$ is Multiplicative
    • 16. The Cycle $L$ of $S^{[2]}$ via Moduli of Stable Sheaves
    • 3. The Variety of Lines on a Cubic Fourfold
    • 17. The Incidence Correspondence $I$
    • 18. The Rational Self-Map $\varphi : F \dashrightarrow F$
    • 19. The Fourier Decomposition for $F$
    • 20. A First Multiplicative Result
    • 21. The Rational Self-Map $\varphi :F\dashrightarrow F$ and the Fourier Decomposition
    • 22. The Fourier Decomposition for $F$ is Multiplicative
    • A. Some Geometry of Cubic Fourfolds
    • B. Rational Maps and Chow Groups
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2402015; 161 pp
MSC: Primary 14; 53;

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.

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