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](https://arxiv.org/abs/1610.07165) 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?



An integration identity on $\mathbb{P}^{n-1}$
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.](https://mathscinet.ams.org/mathscinet-getitem?mr=238226)


