eBook ISBN:  9781470428303 
Product Code:  MEMO/240/1139.E 
List Price:  $90.00 
MAA Member Price:  $81.00 
AMS Member Price:  $54.00 
eBook ISBN:  9781470428303 
Product Code:  MEMO/240/1139.E 
List Price:  $90.00 
MAA Member Price:  $81.00 
AMS Member Price:  $54.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 240; 2015; 161 ppMSC: 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 SelfMap $\varphi : F \dashrightarrow F$

19. The Fourier Decomposition for $F$

20. A First Multiplicative Result

21. The Rational SelfMap $\varphi :F\dashrightarrow F$ 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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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 SelfMap $\varphi : F \dashrightarrow F$

19. The Fourier Decomposition for $F$

20. A First Multiplicative Result

21. The Rational SelfMap $\varphi :F\dashrightarrow F$ and the Fourier Decomposition

22. The Fourier Decomposition for $F$ is Multiplicative

A. Some Geometry of Cubic Fourfolds

B. Rational Maps and Chow Groups