Let $X$ be a smooth manifold, let $E \to X$ be a smooth complex vector bundle of rank $r$, and let $\nabla$ be a smooth connection on $E$ with curvature form $F_\nabla \in \Omega^2(X; \operatorname{End}(E))$. Let $P: \mathfrak{gl}_r(\mathbb{C}) \to \mathbb{C}$ be a polynomial invariant under conjugation by $GL_r(\mathbb{C})$. The Chern-Weil form $P(F_\nabla) \in \Omega^{\mathrm{even}}(X)$ obtained by applying the symmetric polarizations of the homogeneous components of $P$ to $F_\nabla$ is closed:

\begin{align*} d\,P(F_\nabla)=0. \end{align*}