The Fubini--Study metric on an `odd' Grassmannian is rigid

Following the ideas of Gasqui and Goldschmidt, we give an explicit description of the infinitesimal Einstein deformations admitted by the Fubini--Study metric on complex Grassmannians $G_{m}(\mathbb{C}^{n+m})$ with $m,n\geq 2$. The deformations were first shown to exist by Koiso in the 1980s but it has remained an open question as to whether they can be integrated to give genuine deformations of the Fubini--Study metric. We show that when $n+m$ is odd, the answer is no.