An integration identity on $\mathbb{P}^{n-1}$

Question

Let $\omega_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w_1 : \cdots : w_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's paper On real bisectional curvature for Hermitian manifolds they claim the following is well-known: $$\int_{\mathbb{P}^{n-1}} \frac{w_i \overline{w_j} w_k \overline{w_{\ell}}}{\lvert w \rvert^4} \omega_{\text{FS}}^{n-1} = \frac{1}{n(n+1)}(\delta_{ij} \delta_{k \ell} + \delta_{i \ell} \delta_{kj}).$$

Does anyone have a reference for this "well-known fact"? Perhaps is so well-known that nobody knows it?

Answer 1

If I remember correctly, you can find this in the book "Complex Differential Geometry" by Fangyang Zheng in Chapter 7. The analogous result "with real coefficients" (i.e. for real vectors and with $S^{n-1}$ instead of projective space) is originally due to Marcel Berger in Lemma 7.4 here.