Finiteness results for K3 surfaces over arbitrary fields

Bright, Martin; Logan, Adam; Luijk, Ronald van

doi:10.1007/s40879-019-00337-4

Cited by 7 publications

(7 citation statements)

References 17 publications

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“…Using Markman and Yoshioka's argument in [43, Corollary 2.5], the movable cone conjecture for complex hyper-Kähler varieties (proved by Markman [42]) and the boundedness of squares of wall divisors allow Amerik and Verbitsky to conclude the finiteness of the set of birational hyper-Kähler models of a given hyper-Kähler variety over the complex numbers. Over an arbitrary field of characteristic zero, using the method of Bright-Logan-van Luijk [12] for K3 surfaces, Takamatsu [61, Theorem 4.2.7] generalized Markman's work [42] and deduced in the similar way the finiteness of birational hyper-Kähler models: Theorem 3.5 (Takamatsu). Let k be a field of characteristic 0.…”

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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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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“…Let ρ : Aut(X k )⋉R X k → O(Λ X k ) be the natural morphism. Here, we write R X k for the subgroup of O(Λ X k ) generated by reflections by −2 classes as in [BLvL19]. We note that R X k is equal to R X ks which is defined similary, by [BLvL19, Corollary 3.2].…”

Section: Uniform Boundsmentioning

confidence: 99%

“…To prove Theorem 1.0.5, we follow the method given by Bright-Logan-van Luijk [BLvL19], where they proved the cone conjecture for K3 surfaces over non-closed fields of characteristic away from 2.…”

Section: Introductionmentioning

confidence: 99%

On the finiteness of twists of irreducible symplectic varieties

Takamatsu¹

2021

Preprint

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Hyperkähler varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the finiteness of twists of hyperkähler varieties via a fixed finite field extension of characteristic 0. The main ingredient of the proof is the cone conjecture of hyperkähler varieties, which was proved by Markman and Amerik-Verbitsky. To prove the main theorem, we discuss the cone conjecture over nonclosed fields by Bright-Logan-van Luijk's method. We also give an application to the finiteness of derived equivalent twists. Moreover, we discuss the case of K3 surfaces or Enriques surfaces over fields of positive characteristic.

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“…Up to exchanging B 3 and B 4 and B 5 and B 6 , we can suppose that a 1 ≤ b 1 and a 2 ≤ b 2 . One computes that in order for the first rank 4 minor to be 0, we need that the quadruple (a 1 , b 1 , c 1 , d 1 ) is in the following list (0, 8, 8, 0), (1, 7, 7, 1), (2,6,6,2), (3,5,5,3), (4,4,4,4).…”

Section: Constructions Of Surfacesmentioning

confidence: 99%

“…Let X be a K3 surface such that the fundamental domain F X is compact. The automorphism group of X is trivial if and only if it is a smooth quartic surface in P 3 . In that case the possibilities are i) Type S 2 (ρ = 3).…”

Section: Introductionmentioning

confidence: 99%

On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

Roulleau¹

2019

Preprint

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Nikulin [17], [18] and Vinberg [26] proved that there are only a finite number of lattices of rank ≥ 3 that are the Néron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K3 surfaces X, when the fundamental domain FX of their Weyl group in P(NS(X) ⊗ R) is compact. In that case we show that such K3 surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse i.e. if the geometric description we obtained characterizes these surfaces. In 4 cases the description is sufficient, in the 4 other cases there is exactly another one possibility which we study.

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Finiteness results for K3 surfaces over arbitrary fields

Cited by 7 publications

References 17 publications

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

On the finiteness of twists of irreducible symplectic varieties

On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

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