Second order Einstein deformations

We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension 3 turns out to be purely algebraic. One of our main results is the complete and explicit description of integrable to second order infinitesimal Einstein deformations on the complex 2-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $\mathrm{Gr}_{2}(\mathbb{C}^{n+2})$ for $n$ odd is rigid.