  O'Grady constructed a 6-dimensional irreducible holomorphic symplectic
variety by taking a crepant resolution of some moduli space of stable sheaves
on an abelian surface. In this paper, we naturally extend O'Grady's
construction to fields of positive characteristic p greater than 2, called OG6
varieties. We show that a supersingular OG6 variety is unirational, its
rational cohomology group is generated by algebraic classes, and its rational
Chow motive is of Tate type. These results confirm in this case the generalized
Artin--Shioda conjecture, the supersingular Tate conjecture and the
supersingular Bloch conjecture proposed in our previous work, in analogy with
the theory of supersingular K3 surfaces.
