  Let X be the quotient of a smooth projective variety over a field by a finite
group action (in which case we say X is pseudo-smooth), such that the
singularities of X are isolated k-rational points. Let Y be obtained by blowing
up these points on X. Assume further that
  Y is pseudo-smooth, and that the components of the exceptional divisor are
projective spaces. Then, without invoking the theory of finite-dimensional
motives or assuming any of the standard conjectures, we show that a
Chow-Kunneth decomposition on either X or Y gives rise, by means of an explicit
construction, to a Chow-Kunneth decomposition on the other. We use these
constructions to show that various properties (among them, Conjectures of
Murre, and being of Lefschetz type) hold for X if and only if they hold for Y.
The main examples of interest to us are Kummer manifolds. We give several
further applications of our construction to this particular class of examples.
