Let $(X,\omega)$ be a Kähler manifold of complex dimension $n\ge 1$. Let $T_X$ denote the holomorphic tangent bundle, let $K_X=\det(T_X^*)$ denote the canonical bundle, and let $\operatorname{Hol}_x(\omega)$ denote the Levi-Civita holonomy group at a point $x\in X$, viewed through the induced unitary representation on $(T_X)_x$. The following conditions are equivalent:

1. The canonical bundle $K_X$ is holomorphically trivial, and there exists a nowhere-vanishing holomorphic section $\Omega\in H^0(X,K_X)$ whose pointwise norm $|\Omega|_\omega$ is constant on each [connected component](/page/Connected%20Component) of $X$. 2. For every $x\in X$, the holonomy group satisfies $\operatorname{Hol}_x(\omega)\subset SU((T_X)_x,\omega_x)$.