  In this paper, we investigate Murre's conjectures on the structure of
rational Chow groups and exhibit explicit Chow--Kuenneth projectors for some
examples. More precisely, the examples we study are the varieties which have a
nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle
explicit Chow--Kuenneth projectors are obtained which satisfy Murre's
conjectures and the motivic Hard Lefschetz theorem is verified.
