On the rigidity of the complex Grassmannians

We study the integrability to second order of the infinitesimal Einstein deformations of the symmetric metric g on the complex Grassmannian of k-planes inside Cⁿ. By showing the nonvanishing of Koiso's obstruction polynomial, we characterize the infinitesimal deformations that are integrable to second order as an explicit variety inside su(n). In particular we show that g is isolated in the moduli space of Einstein metrics if n is odd.