Derived isogenies and isogenies for abelian surfaces

Li, Zhiyuan; Zou, Haitao

doi:10.48550/arxiv.2108.08710

Cited by 1 publication

(1 citation statement)

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“…The projectivity and smoothness are shown in [17,Prop. 6.9], which relies on classic results in [44] as well as [34] for the construction of moduli spaces of semistable sheaves over arbitrary fields. Geometric irreducibility of 𝑀 𝐴 (𝑣) follows from [29,Thm.…”

Section: Moduli Spaces Over Arbitrary Fieldsmentioning

confidence: 99%

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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