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

Sarah Frei ; Katrina Honigs

Forum of Mathematics, Sigma, Tome 11 (2023)

Résumé

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 $K_A(0,l,s)$.

Détail

DOI : 10.1017/fms.2023.37