Let $X$ be a compact Kähler complex manifold of complex dimension $n$ such that the canonical holomorphic line bundle $K_X=\Lambda^nT_X^*$ is holomorphically trivial, equivalently $X$ admits a nowhere-vanishing holomorphic $n$-form. Here $\mathcal O_X$ denotes the holomorphic structure sheaf of $X$, so holomorphic triviality means $K_X\cong\mathcal O_X$. For a holomorphic vector bundle $E\to X$, let $H^q(X,E)$ denote Dolbeault cohomology, and let $H^{p,q}(X)$ denote Dolbeault cohomology of ordinary $(p,q)$-forms. Let $K$ be the Kuranishi germ parametrizing small complex-structure deformations of $X$, with base point $0\in K$ corresponding to $X$. Then $K$ is smooth at $0$, and its Zariski tangent space is naturally identified with $H^1(X,T_X)$. Equivalently, every first-order deformation class in $H^1(X,T_X)$ is tangent to an actual analytic deformation of the complex structure of $X$.