On the Rigidity of $\mathbb{CP}^{2n}\times \mathbb{CP}^{1}$

We revisit Koiso's original examples of rigid infinitesimally deformable Einstein metrics. We show how to compute Koiso's obstruction to the integrability of the infinitesimal deformations on $\mathbb{CP}^{n}\times \mathbb{CP}^{1}$ using completely elementary complex differential geometry.
Comment: 13 pages