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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?

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See Example 3.1 in volume 2 of Kobayashi–Nomizu; the first Chern class is represented by $-\frac{1}{2\pi i}K$. Tensor with the Kähler form $m-1$ times and integrate over $X$.

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  • $\begingroup$ Thank you for the answer, but I'm not entirely sure how you go from $K_A = \lambda(E)\text{id}$ to the obtained expression for $\lambda(E)$ by the steps you mentioned? Wedging with $\omega^{m-1}$ gives $$\frac{i}{2\pi} K \wedge \omega^{m-1}$$ and integrating over $X$ gives the expression $$\int_X \frac{i}{2\pi} K \wedge \omega^{m-1}$$. @quarto-bendir $\endgroup$
    – Nikolai
    Commented Jun 19 at 17:14
  • $\begingroup$ When you replace $K$ by $\lambda$ you get (up to normalization) the integral of $\omega^m$, which is (up to normalization) the volume. (I think this is sometimes called Wirtinger theorem.) The particular constants $r$ and $(m-1)!$ are just normalizations depending on some conventions of how the various quantities are defined. $\endgroup$
    – Quarto Bendir
    Commented Jun 19 at 20:34

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