Kähler cone on ${\mathbb}C^2$ with an isolated singularity at the origin; and for $t>0$, ${\mathcal}N_t$ is an expanding soliton discovered by Cao [@Cao97]. It is expected that according to most (if not all) of the definitions of a *weak solution of Ricci flow* which are currently being explored (e.g., see [@HN15] and [@Stu16]), the family ${\mathcal}N_t$ will qualify for such a designation. Consequently, in this weak sense, one sees that Ricci flow can carry out an algebraic-geometric blowdown.

As noted above, a pre-condition for a singularity model being generic is that it must be a stable attractor for Ricci flow — regarded as a dynamical system on the space of Riemannian metrics. Stability of ${\mathcal}S^4$ is well established. (In fact, Brendle and Schoen [@BS] show that its basin of attraction includes all $1/4$-pinched metrics.) Stability of generalized cylinders is strongly conjectured but not known for Ricci flow. Stability of the blowdown soliton is also not known, although Máximo’s proof [@Maximo14] shows that arbitrarily small ${\mathrm}U(2)$-invariant Kähler perturbations of the shrinking soliton on ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$ (which was discovered independently by Koiso [@Koiso90] and by Cao [@Cao96]) develop singularities modeled on $({\mathcal}L^2_{-1},h)$. (We remark that prior to Máximo’s results, it was shown in  