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

Abstract

AbstractWe describe the Galois action on the middle$\ell $-adic cohomology of smooth, projective fourfolds$K_A(v)$that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surfaceAwith Mukai vectorv. We show this action is determined by the action on$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]$, which depends onv. This generalizes the analysis carried out by Hassett and Tschinkel over${\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${\mathbb C}$whereAis$(1,3)$-polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of$K_A(0,l,s)$.