1 the Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H , where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t(t − 1)QG/H(t ) for a polynomial QG/H with non-negative integer coefficients. Moreover, we show that QG/H(t ) divides the virtual Poincaré polynomial of every regular embedding of G/H , if H is connected. Introduction and statement of the results One associates to every complex algebraic variety X (possibly singular, or reducible) its virtual Poincaré polynomial PX(t), uniquely determined by the following properties: (i) (additivity) PX(t) = PY (t) + PX−Y (t) for every closed subvariety Y . (ii) If X is smooth and complete, then PX(t) = ∑ m dimH m(X) tm is the usual Poincaré polynomial. Then PX(t) = PY (t) PF (t) for every fibration F → X → Y which is locally trivial for the Zariski topology. Specifically, we have