  The Euler characteristic of Chow varieties of algebraic cycles of a given
degree in complex projective spaces was computed by Blaine Lawson and Stephen
Yau by using holomorphic symmetries of cycles spaces. In this paper we compute
this in a direct and elementary way and generalize this formula to the l-adic
Euler-Poincare characteristic for Chow varieties over any algebraically closed
field. Moreover, the Euler characteristic for Chow varieties with certain group
action is calculated. In particular, we calculate the Euler characteristic of
the space of right quaternionic cycles of a given dimension and degree in
complex projective spaces.
