Existence and asymptotics of a shrinking soliton on the blowup of \mathbb{C}\times\mathbb{P}^1

Let $\operatorname{Bl}_{p}(\mathbb{C} \times \mathbb{P}^{1})$ be the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at a fixed point $p$ of the standard torus action, and let $0<\lambda<1$ be as in the soliton vector field given by

. The real torus action is the one induced by the standard real torus action on $\mathbb{C}\times\mathbb{P}^{1}$. **Existence and asymptotics conjecture.** There \exists a complete shrinking gradient Kähler-Ricci soliton $\omega$ on $\operatorname{Bl}_{p}(\mathbb{C} \times \mathbb{P}^{1})$ invariant under this real torus action, with bounded scalar curvature and with soliton vector field given by

. Moreover, there exists a biholomorphism $\Phi$ of $\operatorname{Bl}_{p}(\mathbb{C} \times \mathbb{P}^{1})$ such that

$$\Phi^*\omega \longrightarrow \frac{i}{2}\partial\bar{\partial}|z|^{\frac{1}{\lambda}}+2\omega_{\mathbb{P}^{1}}$$

at a polynomial rate with respect to the radial coordinate $z$ on the $\mathbb{C}$ factor. This conjecture predicts the existence of the new shrinking gradient Kähler-Ricci soliton identified by the holomorphic classification theorem, together with precise asymptotics; the supplied text does not state a resolution, so the conjecture is recorded as open.