I'm reading Yang-Mills connections and Einstein-Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein-Hermitian connection $A$ by $$K_A = \lambda(E)\text{id}.$$

After this, they state

Using the Chern-Weil formula, we see that the constant $\lambda(E)$ is given by $$\lambda(E) = \frac{2\pi}{r(m-1)!\text{vol}(X)}\int_X c_1(E) \wedge \omega^{m-1}.$$

I have read quite a bit about Chern-Weil theory, but I have never met the so-called "Chern-Weil formula". They do not define it in the notes and I could not find a definition for this online. This certainly isn't the Chern-Weil homomorphism, but something else instead. Anyone here happen to know what is meant by this?

I'm reading Yang–Mills connections and Einstein–Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein–Hermitian connection $A$ by $$K_A = \lambda(E)\mathrm{id}.$$

After this, they state

Using the Chern–Weil formula, we see that the constant $\lambda(E)$ is given by $$\lambda(E) = \frac{2\pi}{r(m-1)!\operatorname{vol}(X)}\int_X c_1(E) \wedge \omega^{m-1}.$$

I have read quite a bit about Chern–Weil theory, but I have never met the so-called "Chern–Weil formula". They do not define it in the notes and I could not find a definition for this online. This certainly isn't the Chern–Weil homomorphism, but something else instead. Does anyone here happen to know what is meant by this?