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

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 ${\mathbb{C}\mathbb{P}}^n\times {\mathbb{C}\mathbb{P}}^1$ using completely elementary complex differential geometry.