Let $M$ be a finite-dimensional smooth manifold, let $E\to M$ be a smooth complex vector bundle of rank $r$, and let $\nabla^E$ be a smooth complex-linear connection on $E$ with curvature $F_{\nabla^E}\in\Omega^2(M;\operatorname{End}(E))$. Let $R\subset\mathbb C$ be a commutative subring, and let
be a formal [power series](/page/Power%20Series) with constant term $1$.
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For each integer $n\ge 0$, let $\Phi_{Q,n}\in R[e_1,\dots,e_r]$ be the unique weighted-[homogeneous polynomial](/page/Homogeneous%20Polynomial) of total Chern degree $n$, where $e_k$ has degree $k$, such that the homogeneous degree-$n$ part of
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\begin{align*}
\prod_{j=1}^r Q(x_j)
\end{align*}
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is equal to $\Phi_{Q,n}(e_1(x),\dots,e_r(x))$, with $e_k(x)$ the $k$-th elementary symmetric polynomial in $x_1,\dots,x_r$. Define
where $c_k(\nabla^E)$ is computed from the normalized curvature $F_{\nabla^E}/(2\pi i)$ by the standard Chern-Weil convention. Since $\dim M<\infty$, both sums are interpreted degreewise and have only finitely many nonzero differential-form components on $M$.
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Then $K_Q(\nabla^E)$ is closed, its de Rham cohomology class is the image of $K_Q(E)$ under the coefficient map