Let $X$ be a complex manifold, let $E \to X$ be a holomorphic vector bundle of rank $r$, and let $h$ be a smooth Hermitian metric on $E$. Let $\nabla^h$ be the Chern connection of $(E,h)$, and let $\Theta_h \in \Omega^2(X;\operatorname{End}(E))$ be its curvature form. Let $I_E \in \Omega^0(X;\operatorname{End}(E))$ denote the identity endomorphism of $E$. For the integer $r$, let $\mathfrak{gl}_r(\mathbb C)$ denote the complex Lie algebra of all $r \times r$ complex matrices, let $GL_r(\mathbb C)$ denote the group of invertible $r \times r$ complex matrices, and let $I_r \in GL_r(\mathbb C)$ denote the identity matrix. Let $\mathcal L^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $[0,1]$ when parameter integrals over metric paths are used. For each $0 \leq k \leq r$, define the Chern-Weil form $c_k(E,h) \in \Omega^{2k}(X;\mathbb C)$ by the identity

\begin{align*} \det\left(I_E + \frac{i}{2\pi}\Theta_h\right) = \sum_{k=0}^{r} c_k(E,h), \end{align*}

where the determinant is computed by the invariant determinant polynomial on endomorphism-valued differential forms, and $c_k(E,h)$ denotes the component of total degree $2k$. Then each $c_k(E,h)$ is closed. Moreover, if $h_0$ and $h_1$ are two smooth Hermitian metrics on $E$, then

\begin{align*} [c_k(E,h_0)] = [c_k(E,h_1)] \in H_{\mathrm{dR}}^{2k}(X;\mathbb C). \end{align*}

With the normalization above, this common de Rham cohomology class is the image of the topological Chern class $c_k(E)$ in complex de Rham cohomology.