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

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