| eBook ISBN: | 978-1-4704-2830-3 |
| Product Code: | MEMO/240/1139.E |
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| AMS Member Price: | $54.00 |
| 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 |
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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.
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Table of Contents
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Chapters
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Introduction
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1. The Fourier Transform for HyperKähler Fourfolds
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1. The Cohomological Fourier Transform
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2. The Fourier Transform on the Chow Groups of HyperKähler Fourfolds
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3. The Fourier Decomposition Is Motivic
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4. First Multiplicative Results
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5. An Application to Symplectic Automorphisms
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6. On the Birational Invariance of the Fourier Decomposition
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7. An Alternate Approach to the Fourier Decomposition on the Chow Ring of Abelian Varieties
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8. Multiplicative Chow–Künneth Decompositions
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9. Algebraicity of $\mathfrak {B}$ for HyperKähler Varieties of $\mathrm {K3}^{[n]}$-type
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2. The Hilbert Scheme $S^{[2]}$
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10. Basics on the Hilbert Scheme of Length-$2$ Subschemes on a Variety $X$
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11. The Incidence Correspondence $I$
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12. Decomposition Results on the Chow Groups of $X^{[2]}$
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13. Multiplicative Chow–Künneth Decomposition for $X^{[2]}$
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14. The Fourier Decomposition for $S^{[2]}$
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15. The Fourier Decomposition for $S^{[2]}$ is Multiplicative
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16. The Cycle $L$ of $S^{[2]}$ via Moduli of Stable Sheaves
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3. The Variety of Lines on a Cubic Fourfold
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17. The Incidence Correspondence $I$
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18. The Rational Self-Map $\varphi : F \dashrightarrow F$
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19. The Fourier Decomposition for $F$
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20. A First Multiplicative Result
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21. The Rational Self-Map $\varphi :F\dashrightarrow F$ and the Fourier Decomposition
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22. The Fourier Decomposition for $F$ is Multiplicative
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A. Some Geometry of Cubic Fourfolds
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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 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
