Let $(X,h)$ be a Hermitian manifold of complex dimension $n$, with the convention that $h$ is $\mathbb{C}$-linear in its first argument and conjugate-linear in its second argument. Let $(U,z)$ be a local holomorphic coordinate chart, with coordinate functions $z_1,\dots,z_n$ and $z_i=x_i+i y_i$. Write
Let $g$ be the associated real Riemannian metric normalized by $g(u,v)=2\operatorname{Re}h(u^{1,0},v^{1,0})$ for real tangent vectors $u,v$, and let the fundamental form $\omega$ be defined by