**Memoirs of the American Mathematical Society**

2015;
161 pp;
Softcover

MSC: Primary 14; 53;

Print ISBN: 978-1-4704-1740-6

Product Code: MEMO/240/1139

List Price: $90.00

AMS Member Price: $54.00

MAA Member Price: $81.00

**Electronic ISBN: 978-1-4704-2830-3
Product Code: MEMO/240/1139.E**

List Price: $90.00

AMS Member Price: $54.00

MAA Member Price: $81.00

# The Fourier Transform for Certain HyperKähler Fourfolds

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*Mingmin Shen; Charles Vial*

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

# Table of Contents

## The Fourier Transform for Certain HyperKahler Fourfolds

- Cover Cover11
- Title page i2
- Introduction 110
- Part 1 . The Fourier Transform for HyperKähler Fourfolds 1524
- Chapter 1. The Cohomological Fourier Transform 1726
- Chapter 2. The Fourier Transform on the Chow Groups of HyperKähler Fourfolds 2534
- Chapter 3. The Fourier Decomposition Is Motivic 3140
- Chapter 4. First Multiplicative Results 3746
- Chapter 5. An Application to Symplectic Automorphisms 4150
- Chapter 6. On the Birational Invariance of the Fourier Decomposition 4352
- Chapter 7. An Alternate Approach to the Fourier Decomposition on the Chow Ring of Abelian Varieties 4756
- Chapter 8. Multiplicative Chow–Künneth Decompositions 5160
- Chapter 9. Algebraicity of 𝔅 for HyperKähler Varieties of 𝔎3^{[𝔫]}-type 6170

- Part 2 . The Hilbert Scheme 𝑆^{[2]} 6978
- Chapter 10. Basics on the Hilbert Scheme of Length-2 Subschemes on a Variety 𝑋 7180
- Chapter 11. The Incidence Correspondence 𝐼 7382
- Chapter 12. Decomposition Results on the Chow Groups of 𝑋^{[2]} 7988
- Chapter 13. Multiplicative Chow–Künneth Decomposition for 𝑋^{[2]} 8594
- Chapter 14. The Fourier Decomposition for 𝑆^{[2]} 97106
- Chapter 15. The Fourier Decomposition for 𝑆^{[2]} is Multiplicative 105114
- Chapter 16. The Cycle 𝐿 of 𝑆^{[2]} via Moduli of Stable Sheaves 113122

- Part 3 . The Variety of Lines on a Cubic Fourfold 115124
- Chapter 17. The Incidence Correspondence 𝐼 119128
- Chapter 18. The Rational Self-Map 𝜙:𝐹\dashrightarrow𝐹 123132
- Chapter 19. The Fourier Decomposition for 𝐹 125134
- Chapter 20. A First Multiplicative Result 129138
- Chapter 21. The Rational Self-Map 𝜙:𝐹\dashrightarrow𝐹 and the Fourier Decomposition 135144
- Chapter 22. The Fourier Decomposition for 𝐹 is Multiplicative 147156
- Appendix A. Some Geometry of Cubic Fourfolds 151160
- Appendix B. Rational Maps and Chow Groups 157166
- References 161170

- Back Cover Back Cover1178