We prove that if A is an abelian variety over ℂ acting algebraically on a complex projective variety X, then the Euler characteristic of X equals the Euler characteristic of the fixed point set XA. We obtain that if A is an abelian variety and X is a principal A-bundle over a projective variety Y, then the Euler characteristic of a Chow variety in X equals either zero or the Euler characteristic of a Chow variety of Y.