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

TL;DR

This paper studies symplectic involutions on symplectic varieties of Kummer type, analyzing their fixed loci and Galois actions on cohomology, revealing new insights into their automorphisms and fixed-point structures.

Contribution

It generalizes Galois action analysis to varieties over number fields and describes the kernel of automorphisms, including fixed loci containing elliptically fibered K3 surfaces.

Findings

Galois action determined by specific subgroup and cohomology actions

Fixed loci of involutions contain K3 surfaces with significant cohomology

Kernel of automorphism map characterized for all varieties of dimension ≥4

Abstract

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 $H^2_{\'et}(A_{\bar{k}},\mathbb{Q}_\ell(1))$ 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 [HT13] over $\mathbb{C}$. 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 $\mathrm{Aut}(K_A(v))\to \mathrm{O}(H^2(K_A(v),\mathbb{Z}))$. 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 loci in fourfolds $K_A(0,l,s)$ over $\mathbb{C}$ where $l$ is a $(1,3)$-polarization, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$.