Let $X$ be a compact Kähler complex manifold of complex dimension $n$ with holomorphically trivial canonical bundle $K_X$. Let $K$ denote the Kuranishi germ parametrizing local complex deformations of $X$, and let $0\in K$ be the point corresponding to the original complex structure on $X$. Then the Bogomolov-Tian-Todorov unobstructedness theorem applies to $X$, so the germ $K$ is smooth at $0$. Moreover the Zariski tangent space $T_0K$ is naturally isomorphic to $H^1(X,T_X)$.