Let $M$ be a smooth manifold of dimension $n$, 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
Let $GL(r,\mathbb C)$ denote the complex [general linear group](/page/General%20Linear%20Group) and let $\mathfrak{gl}(r,\mathbb C)$ denote its [Lie algebra](/page/Lie%20Algebra) of complex $r\times r$ matrices. Use the Chern-Weil normalization
where the sum is truncated by form degree. Then $\operatorname{ch}(E,\nabla^E)$ is closed, and its de Rham cohomology class is independent of the connection $\nabla^E$. Denote this class by
Let $c_k(E)\in H_{\mathrm{dR}}^{2k}(M;\mathbb C)$ denote the $k$-th Chern class, with $c_0(E)=1$ and $c_k(E)=0$ for $k>r$, and let $c(E):=1+c_1(E)+\cdots+c_r(E)$ denote the total Chern class. Let $x_1,\dots,x_r$ be formal degree-two Chern roots, meaning formal variables whose elementary symmetric polynomials satisfy
for $0\le k\le r$. Since $\sum_{j=1}^r e^{x_j}$ is symmetric and is truncated in degrees at most $n$, it denotes the unique finite polynomial in $c_1(E),\dots,c_r(E)$ obtained from the [fundamental theorem of symmetric polynomials](/theorems/5179). With this interpretation, in $H_{\mathrm{dR}}^{\mathrm{even}}(M;\mathbb C)$,