Using a codimension-$1$ algebraic cycle obtained from the Poincare 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 Hyperkahler 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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Höhe: 254 mm
Breite: 178 mm
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ISBN-13
978-1-4704-1740-6 (9781470417406)
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Schweitzer Klassifikation
Mingmin Shen, Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands.
Charles Vial, University of Cambridge, United Kingdom.
Introduction
The Fourier transform for HyperKahler fourfolds
The Cohomological Fourier Transform
The Fourier transform on the Chow groups of HyperKahler fourfolds
The Fourier decomposition is motivic
First multiplicative results
An application to symplectic automorphisms
On the birational invariance of the Fourier decomposition
An alternate approach to the Fourier decomposition on the Chow ring of Abelian varieties
Multiplicative Chow-Kunneth decompositions
Algebraicity of $\mathfrak{B}$ for HyperKahler varieties of $\mathrm{K3}^{[n]}$-type
The Hilbert Scheme $S^{[2]}$
Basics on the Hilbert scheme of Length-$2$ subschemes on a variety $X$
The incidence correspondence $I$
Decomposition results on the Chow groups of $X^{[2]}$
Multiplicative Chow-Kunneth decomposition for $X^{[2]}$
The Fourier decomposition for $S^{[2]}$
The Fourier decomposition for $S^{[2]}$ is multiplicative
The Cycle $L$ of $S^{[2]}$ via moduli of stable sheaves
The variety of lines on a cubic fourfold
The incidence correspondence $I$
The rational self-map $\varphi : F \dashrightarrow F$
The Fourier decomposition for $F$
A first multiplicative result
The rational self-map $\varphi :F\dashrightarrow F$ and the Fourier decomposition
The Fourier decomposition for $F$ is multiplicative
Appendix A. Some geometry of cubic fourfolds
Appendix B. Rational maps and Chow groups
References
