Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces

Given a symplectic involution $\iota$ on a K3 surface $X$, the desingularization $Y$ of $X\mathrm{/}\iota$ is still a K3 surface, which in general has a different Néron–Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen–Sarti involution) and the Picard number is minimal, the Néron–Severi groups of $X$ and $Y$ are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen–Sarti type) whose generic members satisfy $NS(X)\simeq NS(Y)$. Then, we describe the cohomological action of a van Geemen–Sarti involution and we characterize specializations of K3 surfaces with a van Geemen–Sarti involution for which it is still true that $NS(X)\simeq NS(Y)$. There is a 5-dimensional family of K3 surfaces with van Geemen–Sarti involution for which $X\simeq Y$. The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration.