Let $X$ be a complex manifold, let $(E,h)$ be a Hermitian holomorphic vector bundle on $X$, and let $(E^*,h^*)$ be the dual Hermitian holomorphic vector bundle. Let $\nabla^E$ and $\nabla^{E^*}$ denote the Chern connections, and define the curvature forms by
where the transpose is taken on the bundle indices with respect to the natural dual pairing $E^*\times E\to\mathcal O_X$. Equivalently, for every point $x\in X$, every pair of tangent vectors $\xi,\eta\in T_xX$, every $\lambda\in E_x^*$, and every $s\in E_x$,