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

TL;DR

This paper investigates the rigidity of Einstein metrics on complex projective products, specifically $ ext{CP}^{2n} imes ext{CP}^1$, by analyzing Koiso's obstructions through elementary complex differential geometry.

Contribution

It provides a new method to compute Koiso's obstruction for the integrability of infinitesimal deformations on $ ext{CP}^n imes ext{CP}^1$ using elementary techniques.

Findings

Revisits Koiso's examples of rigid infinitesimally deformable Einstein metrics.

Provides an elementary approach to compute obstructions to deformation.

Clarifies the rigidity properties of $ ext{CP}^{2n} imes ext{CP}^1$.

Abstract

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.