  We examine the relationship between the notion of Frobenius splitting and
ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology
groups $H^i(X, W({\mathcal O}_X))$ of a smooth projective Frobenius split
variety are finitely generated over $W(k)$. b) we provide counterexamples to a
question of V. B. Mehta that Frobenius split varieties are ordinary or even
Hodge-Witt. c) a Kummer $K3$ surface associated to an Abelian surface is
$F$-split (ordinary) if and only if the associated Abelian surface is $F$-split
(ordinary). d) for a $K3$-surface defined over a number field, there is a set
of primes of density one in some finite extension of the base field, over which
the surface acquires ordinary reduction. This paper should be read along with
first author's paper `Exotic torsion, Frobenius splitting and the slope
spectral sequence' which is also available in this archive.
