  Let X be a smooth algebraic variety on which a solvable Lie group acts freely
on a dense open orbit. Such varieties include generalized flag varieties, toric
varieties, Bott-Samelson varieties, and many spherical varieties, as well as
equivariant blow-ups of these. We prove that the space of based holomorphic
maps from the Riemann sphere to X approximates in an appropriate sense, both in
homology and in homotopy the space of all continuous based maps from the
2-sphere to X, that is the 2-fold loop space of X.
