Supersingular Tate conjecture for irreducible symplectic varieties of known types: I

Abstract

We prove that for a supersingular irreducible symplectic variety admitting suitable lifting to characteristic zero has Tate Chow motive if it is of deformation type $K3^{[n]}$, OG6 (with Artin invariant $\neq$ 4), or OG10 (with Artin invariant $\neq$ 12), and has supersingular abelian Chow motive if it is of $\mathrm{Kum}^n$-type (with Artin invariant $\neq$ 3). In particular, any product of those irreducible symplectic varieties satisfies the supersingular Tate conjecture for the whole $\ell$-adic or crystalline cohomology ring.