The Fourier Transform for Certain HyperKahler Fourfolds
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The Fourier Transform for Certain HyperKahler Fourfolds: (Memoirs of the American Mathematical Society)

The Fourier Transform for Certain HyperKahler Fourfolds: (Memoirs of the American Mathematical Society)


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About the Book

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.

Table of Contents:
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

About the Author :
Mingmin Shen, Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands. Charles Vial, University of Cambridge, United Kingdom.


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Product Details
  • ISBN-13: 9781470417406
  • Publisher: American Mathematical Society
  • Publisher Imprint: American Mathematical Society
  • Height: 254 mm
  • No of Pages: 161
  • Weight: 319 gr
  • ISBN-10: 1470417405
  • Publisher Date: 30 Apr 2016
  • Binding: Paperback
  • Language: English
  • Series Title: Memoirs of the American Mathematical Society
  • Width: 178 mm


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