  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.
