The Fourier Transform for Certain HyperKähler Fourfolds

Shen, Mingmin; Vial, Charles

doi:10.1090/memo/1139

Cited by 105 publications

(460 citation statements)

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“…For

m > 2

, there is no such “Fourier operator” description of the bigrading

A_{(*)}^{*} true(S^{false[m false]} true)

; the bigrading is defined exclusively by an MCK decomposition. Another point particular to

m = 2

is that (thanks to ) we know that

A_{(j)}^{i} (S^{[2]}) = 0 for all j < 0 .

This vanishing statement is (conjecturally true but) open for

S^{false[m false]}

with

m > 2

.…”

Section: Preliminariesmentioning

confidence: 95%

“…

X = S^{[2]}

is a hyperkähler fourfold). Then the bigrading

A_{(*)}^{*} false(X false)

of Theorem has an interesting alternative description in terms of a Fourier operator on Chow groups . For

m > 2

, there is no such “Fourier operator” description of the bigrading

A_{(*)}^{*} true(S^{false[m false]} true)

; the bigrading is defined exclusively by an MCK decomposition.…”

Section: Preliminariesmentioning

confidence: 99%

“…Any K 3 surface S has an MCK decomposition [, Example 8.17]. Since this property is stable under products [, Theorem 8.6],

S^{m}

also has an MCK decomposition.…”

Section: Preliminariesmentioning

confidence: 99%

“…Any K 3 surface S has an MCK decomposition [, Example 8.17]. Since this property is stable under products [, Theorem 8.6],

S^{m}

also has an MCK decomposition. The following lemma records a basic compatibility between the bigradings on

A^{*} (S^{[m]})

and on

A^{*} false(S^{m} false)

: Lemma Let S be a K 3 surface, and let

X = S^{[m]}

.…”

Section: Preliminariesmentioning

confidence: 99%

“…The assignment

\begin{matrix} true \\ right \begin{matrix} Π_{0}^{S} & : = {(p_{1})}^{*} true(\frac{1}{24} c_{2} (T_{S / B}) true) A^{2} false(scriptS \times_{B} scriptS false), \\ Π_{4}^{S} & : = {(p_{2})}^{*} true(\frac{1}{24} c_{2} (T_{S / B}) true) A^{2} false(scriptS \times_{B} scriptS false), \\ Π_{2}^{S} & : = {normalΔ}_{scriptS} - {normalΠ}_{0}^{scriptS} - {normalΠ}_{4}^{scriptS} \end{matrix} \end{matrix}

defines (by restriction) an MCK decomposition for each fibre, i.e.

{normalΠ}_{j}^{S_{b}} : = {normalΠ}_{j}^{scriptS} |_{S_{b} \times S_{b}} \in A^{2} false(S_{b} \times S_{b} false) false(j = 0, 2, 4 false)

is an MCK decomposition for any

b \in B

[, Example 8.17]. Next, we consider the m ‐fold relative fibre product

S^{m / B}

.…”

Section: Preliminariesmentioning

confidence: 99%

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Algebraic cycles and EPW cubes

Laterveer

2018

Mathematische Nachrichten

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Let $X$ be a hyperk\"ahler variety with an anti-symplectic involution $\iota$. According to Beauville's conjectural "splitting property", the Chow groups of $X$ should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how $\iota$ should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a $19$-dimensional family of hyperk\"ahler sixfolds that are "double EPW cubes" (in the sense of Iliev-Kapustka-Kapustka-Ranestad). This has interesting consequences for the Chow ring of the quotient $X/\iota$, which is an "EPW cube" (in the sense of Iliev-Kapustka-Kapustka-Ranestad).Comment: 32 pages, to appear in Math. Nachrichten, feedback welcom

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“…For

m > 2

, there is no such “Fourier operator” description of the bigrading

A_{(*)}^{*} true(S^{false[m false]} true)

; the bigrading is defined exclusively by an MCK decomposition. Another point particular to

m = 2

is that (thanks to ) we know that

A_{(j)}^{i} (S^{[2]}) = 0 for all j < 0 .

This vanishing statement is (conjecturally true but) open for

S^{false[m false]}

with

m > 2

.…”

Section: Preliminariesmentioning

confidence: 95%

“…

X = S^{[2]}

is a hyperkähler fourfold). Then the bigrading

A_{(*)}^{*} false(X false)

of Theorem has an interesting alternative description in terms of a Fourier operator on Chow groups . For

m > 2

, there is no such “Fourier operator” description of the bigrading

A_{(*)}^{*} true(S^{false[m false]} true)

; the bigrading is defined exclusively by an MCK decomposition.…”

Section: Preliminariesmentioning

confidence: 99%

“…Any K 3 surface S has an MCK decomposition [, Example 8.17]. Since this property is stable under products [, Theorem 8.6],

S^{m}

also has an MCK decomposition.…”

Section: Preliminariesmentioning

confidence: 99%

“…Any K 3 surface S has an MCK decomposition [, Example 8.17]. Since this property is stable under products [, Theorem 8.6],

S^{m}

also has an MCK decomposition. The following lemma records a basic compatibility between the bigradings on

A^{*} (S^{[m]})

and on

A^{*} false(S^{m} false)

: Lemma Let S be a K 3 surface, and let

X = S^{[m]}

.…”

Section: Preliminariesmentioning

confidence: 99%

“…The assignment

\begin{matrix} true \\ right \begin{matrix} Π_{0}^{S} & : = {(p_{1})}^{*} true(\frac{1}{24} c_{2} (T_{S / B}) true) A^{2} false(scriptS \times_{B} scriptS false), \\ Π_{4}^{S} & : = {(p_{2})}^{*} true(\frac{1}{24} c_{2} (T_{S / B}) true) A^{2} false(scriptS \times_{B} scriptS false), \\ Π_{2}^{S} & : = {normalΔ}_{scriptS} - {normalΠ}_{0}^{scriptS} - {normalΠ}_{4}^{scriptS} \end{matrix} \end{matrix}

defines (by restriction) an MCK decomposition for each fibre, i.e.

{normalΠ}_{j}^{S_{b}} : = {normalΠ}_{j}^{scriptS} |_{S_{b} \times S_{b}} \in A^{2} false(S_{b} \times S_{b} false) false(j = 0, 2, 4 false)

is an MCK decomposition for any

b \in B

[, Example 8.17]. Next, we consider the m ‐fold relative fibre product

S^{m / B}

.…”

Section: Preliminariesmentioning

confidence: 99%

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Algebraic cycles and EPW cubes

Laterveer

2018

Mathematische Nachrichten

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Some motivic properties of Gushel–Mukai sixfolds

Bolognesi,

Laterveer

2023

Mathematische Nachrichten

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Gushel–Mukai (GM) sixfolds are an important class of so‐called Fano‐K3 varieties. In this paper, we show that they admit a multiplicative Chow–Künneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double Eisenbud‐Popescu‐Walter (EPW) sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of GM sixfolds.

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Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties

Voisin

2016

Progress in Mathematics

124

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This paper proposes a conjectural picture for the structure of the Chow ring CH * (X) of a (projective) hyper-Kähler variety X, that seems to emerge from the recent papers [9], [23], [24], [25], with emphasis on the Chow group CH 0 (X) of 0-cycles (in this paper, Chow groups will be taken with Q-coefficients). Our motivation is Beauville's conjecture (see [5]) that for such an X, the Bloch-Beilinson filtration has a natural, multiplicative, splitting. This statement is hard to make precise since the Bloch-Beilinson filtration is not known to exist, but for 0-cycles, this means thatwhere the decomposition is given by the action of self-correspondences Γ i of X, and where the group CH 0 (X) i depends only on (i, 0)-forms on X (the correspondence Γ i should act as 0 on H j,0 for j = i, and Id for i = j). We refer to the paragraph 0.1 at the end of this introduction for the axioms of the Bloch-Beilinson filtration and we will refer to it in Section 3 when providing some evidence for our conjectures. Note that a hyper-Kähler variety X has H i,0 (X) = 0 for odd i, so the Bloch-Beilinson filtration F BB has to satisfy F i BB CH 0 (X) = F i+1 BB CH 0 (X) when i is odd. Hence we are only interested in the F 2i BBlevels, which we denote by F ′ i BB . Note also that there are concrete consequences of the Beauville conjecture that can be attacked directly, namely, the 0-th piece CH(X) 0 should map isomorphically via the cycle class map to its image in H * (X, Q) which should be the subalgebra H * (X, Q) alg of algebraic cycle classes, since the Bloch-Beilinson filtration is conjectured to have F 1 BB CH * (X) = CH * (X) hom . Hence there should be a subalgebra of CH * (X, Q) which is isomorphic to the subalgebra H 2 * (X, Q) alg ⊂ H 2 * (X, Q) of algebraic classes. Furthermore, this subalgebra has to contain NS(X) = Pic(X). Thus a concrete subconjecture is the following prediction (cf. This conjecture has been enlarged in [28] to include the Chow-theoretic Chern classes of X, c i (X) := c i (T X ) which should thus be thought as being contained in the 0-th piece of the conjectural Beauville decomposition. Our purpose in this paper is to introduce a new set of classes which should also be put in this 0-th piece, for example, the constant cycles subvarieties of maximal dimension (namely n, with dim X = 2n, because they have to be isotropic, see Section 1) and their higher dimensional generalization, which are algebraically coisotropic. Let us explain the motivation for this, starting from the study of 0-cycles.Based on the case of S [n] where we have the results of [4], [22], [25] that concern the CH 0 group of a K3 surface but will be reinterpreted in a slightly different form in Section 2, we introduce the following decreasing filtration S · on CH 0 (X) for any hyper-Kähler manifold *

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The Fourier Transform for Certain HyperKähler Fourfolds

Cited by 105 publications

References 45 publications

Algebraic cycles and EPW cubes

Algebraic cycles and EPW cubes

Some motivic properties of Gushel–Mukai sixfolds

Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties

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