Derived categories of  surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties

Shen, Junliang; Yin, Qizheng; Zhao, Xiaolei

doi:10.1112/s0010437x19007735

Cited by 22 publications

(25 citation statements)

References 29 publications

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“…Step 2. O'Grady's conjecture [31], which was proven in full generality in [34], implies that both p M and p X (d) are surjective. Hence we can choose a component…”

Section: Algebraically Coisotropic Subvarietiesmentioning

confidence: 97%

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Categories, One-Cycles on Cubic Fourfolds, and the Beauville–voisin Filtration

Shen¹,

Yin

2018

J. Inst. Math. Jussieu

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We explore the connection between K3 categories and 0cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative K3 category associated to a nonsingular cubic 4-fold.By introducing a filtration on the CH1-group of a cubic 4-fold Y , we conjecture a sheaf/cycle correspondence for the associated K3 category AY . This is a noncommutative analog of O'Grady's conjecture concerning derived categories of K3 surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.Our method provides systematic constructions of (a) the Beauville-Voisin filtration on the CH0-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in AY .

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“…Step 2. O'Grady's conjecture [31], which was proven in full generality in [34], implies that both p M and p X (d) are surjective. Hence we can choose a component…”

Section: Algebraically Coisotropic Subvarietiesmentioning

confidence: 97%

“…The following generalized version of O'Grady's conjecture [31] is proven in [34], based on earlier results of Huybrechts, O'Grady, and Voisin in [15,31,40]. [34] for further details.…”

Section: O'grady's Conjecturementioning

confidence: 99%

Categories, One-Cycles on Cubic Fourfolds, and the Beauville–voisin Filtration

Shen¹,

Yin

2018

J. Inst. Math. Jussieu

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“…From (5.7) we get a map t 2 (S)(1) → t(X) such that A 3 (t 2 (S)(1)) = A 0 (S) 0 → A 3 (t(X) = A 1 (X) hom is surjective and we are left to show that it is an isomorphism. By [SYZ,Theorem 3.6] there is an isomorphism A 0 (S) 0 → A 0 (F ) 2 , where F = F (X), that, together with the isomorphism A 0 (F ) 2 ≃ A 1 (X) hom in Theorem 2.5, gives the isomorphism A 0 (S) 0 ≃ A 3 (t(X)) = A 1 (X) hom .…”

Section: Cubic Fourfolds Fibered Over a Planementioning

confidence: 99%

“…The map (Γ) * ) in (5.7) gives a map (Γ) * : t 2 (S)(1) → t(X), such that the associated map on Chow groups A 0 (S) 0 → A 1 (X) hom in 5.9 is surjective. In order to show that is also injective and hence the map (Γ) * gives an isomorphism of motives, we apply the same argument as in the proof of [SYZ,Thm. 3.6].…”

Section: 2mentioning

confidence: 99%

The transcendental motive of a cubic fourfold

Bolognesi

Pedrini

2020

Journal of Pure and Applied Algebra

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In this note we introduce the transcendental part t(X) of the motive of a cubic fourfold X and prove that it is isomorphic to the (twisted) transcendental part h tr 2 (F (X)) in a suitable Chow-Künneth decomposition for the motive of the Fano variety of lines F (X). Then we prove that t(X) is isomorphic to the Prym motive associated to the surface S l ⊂ F (X) of lines meeting a general line l. If X is a special cubic fourfold in the sense of Hodge theory, and F (X) ≃ S [2] , with S a K3, then we show that t(X) ≃ t 2 (S)(1), where t 2 (S) is the transcendental motive. Therefore the motive h(X) is finite dimensional if and only if S has a finite dimensional motive. If X is very general then t(X) cannot be isomorphic to the (twisted) transcendental motive of a surface. We relate the existence of an isomorphism t(X) ≃ t 2 (S)(1) to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we consider the case of cubic fourfolds X admitting a fibration over P 2 , whose fibers are either quadrics or del Pezzo surfaces of degree 6, and prove the isomorphism t 2 (S)(1) ≃ t(X), with S a K3 surface.

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“…We first observe that the image of the moduli space of stable objects on an Enriques surface is a constant cycle Lagrangian. Shen, Yin, and Zhao [22] studied the group of zerocycles on moduli spaces of stable objects on K3 surfaces. They formulated a conjecture, which was later proven by the third author and Marian [17].…”

Section: Moduli Spaces Of Stable Objects On Enriques Surfacesmentioning

confidence: 99%

Birational geometry of moduli spaces of stable objects on Enriques surfaces

Thorsten

2020

Sel. Math. New Ser.

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Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes and that under an extra assumption every minimal model can be described as a moduli space. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.

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Derived categories of surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties

Cited by 22 publications

References 29 publications

Categories, One-Cycles on Cubic Fourfolds, and the Beauville–voisin Filtration

Categories, One-Cycles on Cubic Fourfolds, and the Beauville–voisin Filtration

The transcendental motive of a cubic fourfold

Birational geometry of moduli spaces of stable objects on Enriques surfaces

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