On the finiteness of twists of irreducible symplectic varieties

Takamatsu, Teppei

doi:10.48550/arxiv.2106.11651

Cited by 3 publications

(5 citation statements)

References 16 publications

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“…In the case of K3 surfaces, such a statement can be justified by the action of Weyl groups on Picard lattices, but for general irreducible symplectic varieties, we need additional arguments since there may exist birationally equivalent non-isomorphic irreducible symplectic varieties. Such a problem is related to the Cone conjecture for irreducible symplectic varieties, and actually already discussed in [Tak21].…”

Section: Introductionmentioning

confidence: 93%

On the Shafarevich conjecture for irreducible symplectic varieties

Takamatsu¹

2022

Preprint

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Section: Introductionmentioning

confidence: 93%

On the Shafarevich conjecture for irreducible symplectic varieties

Takamatsu¹

2022

Preprint

Self Cite

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“…According to Proposition 2.12, the cohomological Shafarevich condition can not distinguish between birationally isomorphic elements in Shaf hom M (F, R). This suggests our approach towards the Shafarevich conjecture: on one hand, we need the finiteness of isomorphism classes within a birational isomorphism class (this is guaranteed by a result of Takamatsu [61], recalled below as Theorem 3.5); on the other hand, we study the finiteness of birational isomorphism classes in various Shafarevich sets, which is the main goal of the paper, accomplished in Section 7.3).…”

Section: 3mentioning

confidence: 99%

“…where f runs through all k-birational isomorphisms between X and other hyper-Kähler varieties over k. It is known that the closure of BA(X) is the movable cone Mov(X) (cf. [20,61]). Definition 3.1.…”

Section: Wall Divisors On Hyper-k äHler Varietiesmentioning

confidence: 99%

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Fu¹,

Li²,

Zou³

2022

Preprint

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Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher dimensional analogues of K3 surfaces, Y. André has verified the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach is a uniform Kuga-Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest.

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“…where f runs through all k-birational isomorphisms from X to other hyper-Kähler varieties over k. It is known that the closure of BA(X) is the movable cone Mov(X) (see [27,79]). Definition 2.11.…”

Section: Polarization In Familymentioning

confidence: 99%

“…In other words, the unramifiedness condition can not distinguish between birationally isomorphic members in Shaf hom M (F, R). This suggests our approach towards the Shafarevich conjecture: on one hand, we have the finiteness of isomorphism classes within a birational isomorphism class (this is guaranteed by [79], recalled above as Theorem 2.15); on the other hand, we study the finiteness of birational isomorphism classes in various Shafarevich sets, which is the main goal of the paper, accomplished in Section 7).…”

Section: Various Shafarevich Sets Of Hyper-k äHler Varietiesmentioning

confidence: 99%

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Fu¹,

Li²,

Takamatsu³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher-dimensional analogues of K3 surfaces, Y. André proved the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. We also discuss the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition by the unramifiedness of the cohomology, where our results are subject to certain necessary assumption on the faithfulness of the action of the automorphism group on cohomology. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach to these results is a uniform Kuga-Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest.

show abstract

On the finiteness of twists of irreducible symplectic varieties

Cited by 3 publications

References 16 publications

On the Shafarevich conjecture for irreducible symplectic varieties

On the Shafarevich conjecture for irreducible symplectic varieties

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

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