Let \(M = \mathbb C P^2 \# \overline{\mathbb C P^2}\) be the blow-up of \(\mathbb C P^2\) at one point. It is known that \(M\) is a toric Fano manifold with non-reductive automorphism group and hence with no Kähler-Einstein metric. So, if one follows the continuity method to solve the Kähler-Einstein equation on \(M\), i.e., if one considers a Kähler metric \(g\) and the 1-parameter family of complex Monge-Ampère equations
\[ \det\left(g_{i \bar j} + \phi_{i \bar j}\right) = \det\left(g_{i \bar j}\right) e^{h - \lambda \phi} \eqno(\star)_\lambda \]
(here \(h\) is the Ricci potential of \(g\)), the set of \(\lambda\)s for which equation \((\star)_\lambda\) admits solutions cannot include the value \(\lambda = 1\). Indeed, it is known that, if \(g\) is a Calabi symmetric metric, the Kähler metrics \(g_\lambda\), given by solutions of \((\star)_\lambda\), blow up for \(\lambda \to \frac{6}{7}\).
In this paper, the authors study the geometry of the limit space determined by a sequence of Kähler metrics \(g_{\lambda_i}\) converging to a singular metric \(g_\infty\) when \(\lambda_i \to \frac{6}{7}\). More precisely, they prove the existence of a sequence \(g_{\lambda_i}\), which converges in the Cheeger-Gromov sense to a singular metric \(g_{\infty}\) such that: (a) it is smooth on \(M \setminus E_2\), where \(E_2\) is the infinity section of the ruled surface \(M\); (b) it has conical singularities on \(E_2\) with the same conical angle \(\frac{10 \pi}{7}\) along one direction; (c) its Ricci curvature is bounded. Notice that \(g_\infty\) is not a Kähler-Ricci soliton, in contrast to the property of the Cheeger-Gromov limits of Kähler-Ricci flows on toric Fano manifolds, proved by the second author.