Algebraic Cycles on a Generalized Kummer Variety

Ze, XU

doi:10.1093/imrn/rnw266

Cited by 9 publications

(10 citation statements)

References 17 publications

(14 reference statements)

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“…• Note that the correspondence in [19] which is used in §5 for Case (A) is a special case of Theorem 6.1. • Theorem 6.1 is used in [62] to deduce a motivic decomposition of generalized Kummer varieties equivalent to the Corollary 6.3 below.…”

Section: Remarks 62mentioning

confidence: 99%

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Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties

Fu¹,

Tian²,

Vial³

2019

Geom. Topol.

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Given a smooth projective variety M endowed with a faithful action of a finite group G, following Jarvis-Kaufmann-Kimura [36] and Fantechi-Göttsche [26], we define the orbifold motive (or Chen-Ruan motive) of the quotient stack [M/G] as an algebra object in the category of Chow motives. Inspired by Ruan [51], one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture (CHRC). We prove this motivic version, as well as its K-theoretic analogue conjectured in [36], in two situations related to an abelian surface A and a positive integer n. Case (A) concerns Hilbert schemes of points of A : the Chow motive of A [n] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A n /S n ]. Case (B) for generalized Kummer varieties : the Chow motive of the generalized Kummer variety K n (A) is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A n+1is the kernel abelian variety of the summation map A n+1 → A. As a byproduct, we prove the original Cohomological HyperKähler Resolution Conjecture for generalized Kummer varieties.As an application, we provide multiplicative Chow-Künneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch-Beilinson-Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville [10]. Finally, as another application, we prove that over a non-empty Zariski open subset of the base, there exists a decomposition isomorphism Rπ * Q ≃ ⊕R i π * Q[−i] in D b c (B) which is compatible with the cup-products on both sides, where π : K n (A) → B is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces A → B.Meta-conjecture 1.2 (MHRC). Let X be a smooth proper complex Deligne-Mumford stack with underlying coarse moduli space X being a (singular) symplectic variety. If there is a symplectic resolution Y → X, then we have an isomorphism h(Y) ≃ h orb (X) of commutative algebra objects in CHM C , hence in particular an isomorphism of graded C-algebras : CH * (Y) C ≃ CH * orb (X) C .See Definition 3.1 for generalities on symplectic singularities and symplectic resolutions. The reason why it is only a meta-conjecture is that the definition of orbifold Chow motive for a smooth proper Deligne-Mumford stack in general is not available in the literature and we will not develop the theory in this generality in this paper (see however Remark 2.10). From now on, let us restrict ourselves to the case where the Deligne-Mumford stack in question is of the form of a global quotient X = [M/G], where M is a smooth projective variety with a faithful action of a finite group G, in which case we will define the orbifold Chow motive h orb (X) in a very explicit way in Definition 2.5.The Motiv...

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Section: Remarks 62mentioning

confidence: 99%

“…Let us start with the existence of a self-dual multiplicative Chow-Künneth decomposition : [20] as explained in §6.1 (see Corollary 6.3) and is explicitly written down by Z. Xu [62].…”

Section: Candidate Decompositions In Case (mentioning

confidence: 99%

Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties

Fu¹,

Tian²,

Vial³

2019

Geom. Topol.

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“…where B is a disjoint union of abelian varieties and where Γ is a correspondence between B and X m . Examples of varieties motivated by an abelian surface include generalized Kummer varieties (see [57], and also [22,Corollary 6.3]). In particular, the following theorem applies to generalized Kummer varieties.…”

Section: 4mentioning

confidence: 99%

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for Abelian varieties

Vial¹

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let γ be a correspondence between two abelian varieties A and B that can be written as a linear combination of products of symmetric divisors. Assume that A is isogenous to the product of an abelian variety of totally real type with the power of an abelian surface. We show that γ satisfies the conclusion of Bloch's conjecture. A key ingredient consists in establishing a strong form of the generalized Hodge conjecture for Hodge sub-structures of the cohomology of A that arise as sub-representations of the Lefschetz group of A. As a by-product of our method, we use a strong form of the generalized Hodge conjecture established for powers of abelian surfaces to show that every finite-order symplectic automorphism of a generalized Kummer variety acts as the identity on the zero-cycles. γ * CH 0 (X) = 0. The Bloch-Srinivas argument [14] implies that γ * H * (X, Q) is supported on a divisor, which in turn implies that γ * H i,0 (X) = 0 for all integers i. The Bloch conjecture stipulates that, conversely, should γ * H i,0 (X) vanish for all integers i, then γ acts nilpotently on CH 0 (X). (In fact, the conjecture predicts that γ should act as zero on the graded pieces of the conjectural Bloch-Beilinson filtration on CH 0 (X).)More generally, if γ * CH r (X) = 0 for all r < n, then the Bloch-Srinivas argument implies that γ * H * (X, Q) is supported on a subscheme of codimension n, which in turn implies that γ * H i,j (X) = 0 for all integers i and j < n. The generalized Bloch conjecture is the following converse assertion :Conjecture 1 (Generalized Bloch conjecture). Let X be a smooth projective complex variety of dimension d, and let γ ∈ CH d (X × X) be a correspondence. Suppose that γ * H i,j (X) = 0 for all j < n, or, equivalently in terms of the Hodge coniveau filtration, γ * H * (X, Q) ⊆ N n H H * (X, Q). Then γ * acts nilpotently on CH r (X) for all r < n.The conjecture is wide open, but has notably been established for surfaces with H 2,0 = 0 not of general type [13], for certain surfaces with H 2,0 = 0 of general type [52,55], and for finite-order symplectic automorphisms of K3 surfaces [54,28].Conjecture 1 follows from the combination of (a) the generalized Hodge conjecture (for smooth projective varieties, and not just for X) and (b) the existence of the conjectural Bloch-Beilinson filtration. Indeed, if γ * H i,j (X) = 0 for all j < n, then the generalized Hodge conjecture for X implies that γ * H * (X, Q) is supported on a closed subscheme X of codimension n. By [7], the

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“…Next, we observe that (as X is hyperkähler) H 3 (X, O X ) = 0. Since the generalized Hodge conjecture is known to hold for self-products of abelian surfaces [1, 7.2.2], [2, 8.1(2)], and generalized Kummer varieties are motivated by abelian surfaces in the sense of [3], the generalized Hodge conjecture is true for generalized Kummer varieties (for the usual Hodge conjecture, this was noted in [22,Theorem 3.3]). In particular, H 3 (X) is supported on a divisor D ⊂ X, and H 2n−3 (X) is supported on a 2-dimensional subvariety S ⊂ X.…”

Section: Generalized Kummer Varietiesmentioning

confidence: 99%

“…Here, we have used the following lemma. (The lemma applies to our set-up, because generalized Kummer varieties have finite-dimensional motive [22], [9].) Lemma 3.3.…”

Section: Generalized Kummer Varietiesmentioning

confidence: 99%