Supersingular Irreducible Symplectic Varieties

Fu, Lie; Li, Zhiyuan

doi:10.1007/978-3-030-75421-1_7

Cited by 7 publications

(6 citation statements)

References 93 publications

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“…In [16] the authors exhibit an abelian surface A defined over a number field k where Q ℓ [ 𝐴 [3]] and Q ℓ [ Â [3]] are not isomorphic as Galois modules over k. Remark 5.9. If A is an abelian surface as in the proof of Corollary 5.8, any derived equivalence between 𝐾 2 ( 𝐴) and 𝐾 2 ( Â) would have to be defined over a field larger than k. Moreover, the kernel of such a derived equivalence could not be constructed out of only universal bundles, since such bundles would naturally be defined over k, and the derived equivalence would descend.…”

Section: Derived Equivalence Of Fourfolds Of Kummer Typementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In forthcoming work [16] on Galois actions on torsion subgroups of abelian surfaces, examples of such abelian surfaces are constructed. Intriguingly, this corollary shows that if 𝐾 2 ( 𝐴) and 𝐾 2 ( Â) are derived equivalent after base change to C, then the kernel of the Fourier-Mukai transform cannot be given by naturally associated bundles that would descend to the field of definition for A. Corollary 1.2 complements the recent work of Magni [37], which provides a sufficient condition for the existence of such equivalences over algebraically closed fields of characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

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Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Frei

Honigs

2023

Forum of Mathematics, Sigma

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We describe the Galois action on the middle $\ell $ -adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface A with Mukai vector v. We show this action is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat {A})[3]$ , which depends on v. This generalizes the analysis carried out by Hassett and Tschinkel over ${\mathbb C}$ [21]. As a consequence, over number fields, we give a condition under which $K_2(A)$ and $K_2(\hat {A})$ are not derived equivalent. The points of $G_A(v)$ correspond to involutions of $K_A(v)$ . Over ${\mathbb C}$ , they are known to be symplectic and contained in the kernel of the map $\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$ . We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$ . When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds $K_A(0,l,s)$ over ${\mathbb C}$ where A is $(1,3)$ -polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$ .

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Section: Derived Equivalence Of Fourfolds Of Kummer Typementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Frei

Honigs

2023

Forum of Mathematics, Sigma

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“…If defined over C, the variety K A (v) is a hyperkähler variety of dimension v 2 − 2 and is deformation equivalent to the generalized Kummer variety K n (A) ∼ = K A (1, 0, −n − 1) where n := v 2 2 − 1, which is given by the fiber over 0 of the summation map acting on the Hilbert scheme of length-(n + 1) points on A. Following Fu and Li [FL21], who study these varieties over other fields, we call the K A (v) symplectic varieties (see Proposition 2.5). There are four known deformation types of hyperkähler varieties: K3 [n] type, Kummer type (or Kum n -type), and the two sporadic examples of O'Grady [O'G99, O'G03].…”

Section: Introductionmentioning

confidence: 99%

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Frei¹,

Honigs²

2022

Preprint

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We describe the Galois action in characteristic zero on the middle ℓ-adic cohomology of smooth, projective fourfolds KA(v) that occur as a fiber of the Albanese morphism on moduli spaces of geometrically stable sheaves on an abelian surface A with Mukai vector v. We show that this action is completely determined by the action on H 2 ét (Ak, Q ℓ (1)) and on a subgroup GA(v) (A × Â)[3] which depends on v. This generalizes the analysis carried out by Hassett and Tschinkel [HT13] over C. As a consequence, we prove that over number fields, K2(A) and K2( Â) are not in general derived equivalent.An essential step in the proof is providing an explicit description of the elements of the kernel of ν :for KA(v) over C; we give this description for all varieties KA(v) of dimension at least 4.Over arbitrary fields, the points of GA(v) correspond to symplectic involutions contained in ker ν, and their fixed-point loci contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. While the group structure of ker ν was previously known, the explicit description is required to study the Galois action on the fixed-point loci.It was previously shown that the fixed-point locus of such a symplectic involution is a K3 surface along with 36 isolated points. We examine the fixed-point locus on fourfolds KA(0, l, s) over C where A is (1, 3)polarized, finding the K3 surface elliptically fibered under a Lagrangian fibration of KA(0, l, s).

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“…The study of such varieties is still in its early stages: besides K3 surfaces, it is not clear how to define these objects (but a possible definition is proposed in [FL18]). Despite this, certain higher dimensional moduli spaces of sheaves on K3 surfaces play a key role in Charles's proof of the Tate conjecture for K3 surfaces [Cha16] over finite fields; other examples of hyper-Kähler varieties over finite fields can be obtained from moduli spaces of sheaves on abelian surfaces [FL18]. All these varieties can be lifted to some hyper-Kähler variety in characteristic 0.…”

Section: Introductionmentioning

confidence: 99%

Galois representations on the cohomology of hyper-Kähler varieties

Floccari

2020

Preprint

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We show that the André motive of a hyper-Kähler variety X over a field K ⊂ C with b 2 (X) > 3 is governed by its component in degree 2. More precisely we prove that if X 1 and X 2 are deformation equivalent hyper-Kähler varieties with b 2 (X i ) > 3 and if there exists a Hodge isometry f :the André motives of X 1 and X 2 are isomorphic after a finite extension of K. As a consequence, the Galois representations on the étale cohomology of X 1 and 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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Supersingular Irreducible Symplectic Varieties

Cited by 7 publications

References 93 publications

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Galois representations on the cohomology of hyper-Kähler varieties

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