Deformation Principle and André motives of Projective Hyperkähler Manifolds

Soldatenkov, Andrey

doi:10.1093/imrn/rnab108

Cited by 3 publications

(11 citation statements)

References 26 publications

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“…The combination of Corollary 1.11 and Theorem 1.14 (plus the fact that two deformation equivalent hyper-Kähler varieties can be connected by algebraic families) implies that the abelianity of the André motive of hyper-Kähler varieties is a deformation invariant property (Corollary 7.2(i)). When finalizing the paper, we discovered the recent update of Soldatenkov's preprint [76], where he also obtained this result, as well as Corollary 1.16(i), except for the O'Grady-10 case. We attribute the overlap to him.…”

Section: Introductionmentioning

confidence: 65%

“…The item (i) on the abelianity of André motive is proved for K3 [n] -type by Schlickewei [74], for Kum n -type and OG6-type in the recent work of Soldatenkov [76].…”

Section: Introductionmentioning

confidence: 97%

“…Remark 1.18 (Relation to [76]). The combination of Corollary 1.11 and Theorem 1.14 (plus the fact that two deformation equivalent hyper-Kähler varieties can be connected by algebraic families) implies that the abelianity of the André motive of hyper-Kähler varieties is a deformation invariant property (Corollary 7.2(i)).…”

Section: Introductionmentioning

confidence: 99%

“…We attribute the overlap to him. The proofs and points of view are somewhat different: [76] makes a detailed study of the Kuga-Satake construction in families, while our argument does not involve the Kuga-Satake construction when the odd cohomology is trivial, but relies on André's theorem [3] on the abelianity of H 2 . As a bonus of emphasizing the usage of defect groups in our study, on the one hand, there seems to be some promising approaches mentioned in Remark 1.13 to show the abelianity of the André motive of hyper-Kähler varieties in general; on the other hand, even if Conjecture 1.12 (hence the abelianity) turned out to fail for some deformation family of hyper-Kähler varieties, our method can still control their André motives by its degree-2 part together with informations on the André motive of one given member in that family, see Corollary 7.2 (ii), (iii).…”

Section: Introductionmentioning

confidence: 99%

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On the motive of O'Grady's ten-dimensional hyper-Kähler varieties

Floccari,

Fu,

Zhang

2019

Preprint

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We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O'Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O'Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford-Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford-Tate conjecture holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

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Section: Introductionmentioning

confidence: 65%

“…The item (i) on the abelianity of André motive is proved for K3 [n] -type by Schlickewei [74], for Kum n -type and OG6-type in the recent work of Soldatenkov [76].…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the motive of O'Grady's ten-dimensional hyper-Kähler varieties

Floccari,

Fu,

Zhang

2019

Preprint

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“…In similar spirit, Soldatenkov [28] shows that, under the assumption that b 2 > 3, the varieties X 1 and X 2 can be joined via smooth and proper (but not necessarily projective) families over curves; however, the total space of such a family is not an algebraic variety in general.…”

Section: Remark 52mentioning

confidence: 91%

Galois representations on the cohomology of hyper-Kähler varieties

Floccari

2022

Math. Z.

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We show that the André motive of a hyper-Kähler variety X over a field $$K \subset {\mathbb {C}}$$ K ⊂ C with $$b_2(X)>6$$ b 2 ( X ) > 6 is governed by its component in degree 2. More precisely, we prove that if $$X_1$$ X 1 and $$X_2$$ X 2 are deformation equivalent hyper-Kähler varieties with $$b_2(X_i)>6$$ b 2 ( X i ) > 6 and if there exists a Hodge isometry $$f:H^2(X_1,{\mathbb {Q}})\rightarrow H^2(X_2,{\mathbb {Q}})$$ f : H 2 ( X 1 , Q ) → H 2 ( X 2 , Q ) , then the André motives of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.

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On the motive of O'Grady's six dimensional hyper-K\"{a}hler varieties

Floccari¹

2023

Épijournal De Géométrie Algébrique

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We prove that the rational Chow motive of a six dimensional hyper-K\"{a}hler variety obtained as symplectic resolution of O'Grady type of a singular moduli space of semistable sheaves on an abelian surface $A$ belongs to the tensor category of motives generated by the motive of $A$. We in fact give a formula for the rational Chow motive of such a variety in terms of that of the surface. As a consequence, the conjectures of Hodge and Tate hold for many hyper-K\"{a}hler varieties of OG6-type.

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Deformation Principle and André motives of Projective Hyperkähler Manifolds

Cited by 3 publications

References 26 publications

On the motive of O'Grady's ten-dimensional hyper-Kähler varieties

On the motive of O'Grady's ten-dimensional hyper-Kähler varieties

Galois representations on the cohomology of hyper-Kähler varieties

On the motive of O'Grady's six dimensional hyper-K\"{a}hler varieties

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