Rational homotopy stability for the spaces of rational maps

Let ${\text{Hol}}_{x_0}^{\mathbf{n}}(\text{C}{\mathrm{\P }}^1,X)$ be the space of based holomorphic maps of degree $\mathbf{n}$ from $\text{C}{\mathrm{\P }}^1$ into a simply connected algebraic variety $X$. Under some condition we prove that the map $\map \Hol_{x_0}^{{\bf n}} (\C\P^1, X). \Hol_{x_0}^{d{\bf n}} (\C\P^1, X).$obtainedbycompositing$f \in \Hol_{x_0}^{{\bf n}} (\C\P^1, X)$with$g(z)=z^d, z \in \C\P^1$ induces rational homotopy equivalence up to some dimension, which tends to infinity as the degree grows.