Taking quotient by a unipotent group induces a homotopy equivalence

Let U be a unipotent group over the field of complex numbers C, acting on a complex algebraic variety X. Assume that there exists a surjective morphism of complex algebraic varieties f: X --> Y whose fibres are orbits of U. We show that if X and Y are smooth and all orbits of U in X have the same dimension, then the induced map on C-points X© --> Y© is a homotopy equivalence. Moreover, if U, X, Y, and f are defined over the field of real numbers R, then the induced map on R-points X® --> Y® is surjective and induces homotopy equivalences on connected components.