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The Fourier transform for certain hyperkähler fourfolds. (English) Zbl 1386.14025

Mem. Am. Math. Soc. 1139, viii, 168 p. (2016).
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, in book gived 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. Further was 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.
The manuscript is divided into three parts. In Chapter 1 the Beauville-Bogomolov class \(\mathfrak{B}\) is introduced and establish in Proposition 1.3 the quadratic equation. The core of Part 1 consists in Theorems 2.2 & 2.4 and Theorem 3.3, where was considered a hyperKähler variety \(F\) of \(K3\) -type endowed with a cycle \(L \in \mathrm{CH}^2 (F \times F)\) representing the Beauville-Bogomolov class \(\mathfrak{B}\), and show that the conclusion of Theorem 2 holds for \(F\) and that the resulting Fourier decomposition on the Chow groups of \(F\) is in fact induced by a Chow-Künneth decomposition. In Chapter 6 was shown that the multiplicativity property of the Fourier decomposition boils down to intersection-theoretic properties of the cycle \(L\) and deduced that the Fourier decomposition is a birational invariant for hyperKähler varieties of \(K3^{[2]}\)-type. This approach is used in Chapter 7 to give new insight on the theory of algebraic cycles on abelian varieties by showing how Beauville’s Fourier decomposition theorem is a direct consequence of a recent theorem of P. O’Sullivan [J. Reine Angew. Math. 654, 1–81 (2011; Zbl 1258.14006)]. In Section 8 introduces the notion of multiplicative Chow-Künneth decomposition and its relevance is discussed. Part 1 ends with Chapter 9 where a proof of the algebraicity of the Beauville-Bogomolov class \(\mathfrak{B}\) is given for hyperKähler varieties of \(K3^{[n]}\)-type.
Part 2 and Part 3 are devoted to proving Theorem 1, Theorem 2 and Theorem 3 for the Hilbert scheme of length-2 subschemes on a \(K3\) surface and for the variety of lines on a cubic fourfold, respectively. In both cases, the strategy for proving Theorem 1 and Theorem 2 consists in first studying the incidence correspondence \(I\) and its intersection-theoretic properties, and then in constructing a cycle \(L \in \mathrm{CH}^2 (F \times F)\) very close to \(I\) representing the Beauville-Bogomolov class satisfying some known hypotheses.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14J35 \(4\)-folds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry

Citations:

Zbl 1258.14006
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References:

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