  A smooth complex quasi-affine algebraic variety $Y$ is flexible if its
special group $\SAut (Y)$ of automorphisms (generated by the elements of
one-dimensional unipotent subgroups of $\Aut (Y)$) acts transitively on $Y$. An
irreducible algebraic manifold $X$ is locally stably flexible if it is the
union $\bigcup X_i$ of a finite number of Zariski open sets, each $X_i$ being
quasi-affine, so that there is a positive integer $N$ for which $X_i\times
\mathbb{C}^N$ is flexible for every $i$. The main result of this paper is that
the blowup of a locally stably flexible manifold at a smooth algebraic
submanifold (not necessarily equi-dimensional or connected) is subelliptic, and
hence Oka. This result is proven as a corollary of some general results
concerning the so-called $k$-flexible manifolds.
