Let X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.

Let X be an irreducible projective variety of dimension n in a projective space and let x be a point of X. Denote by Curvesd(X, x) the space of curves of degree d lying on X and passing through x. We will show that the number of components of Curvesd(X, x) for any smooth point x outside a subvariety of codimension $\geq 2$ is bounded by a number depending only on n and d. An effective bound is given. A key ingredient of the proof is an argument from Ein, Küchle and Lazarsfeld's work on Seshadri numbers.