I'm trying to find the scalar curvature/Ricci scalar $R$ of $\mathbb{CP}^n$ under the Fubini-Study metric, $g_{\mu \bar{\nu}}^{\textrm{FS}}$.
As $\mathbb{CP}^n$ is a Kahler manifold, we know that the Ricci form is given by:
\begin{align} \mathcal{R} &= -i\partial \bar{\partial} \log \sqrt{\textrm{det} \; g_{ab}} \\ &= i(n+1) \, g_{\mu \bar{\nu}}^{\textrm{FS}} dz^{\mu} \wedge d\bar{z}^{\nu} \end{align}
(Where $g_{ab}$ is the full matrix and $g_{\mu \bar{\nu}}$ is the part with the holomorphic and anti-holomorphic indices.) The coefficients of the Ricci form coincide with the Ricci scalar, (modulo some factors of $i$?), so $\mathcal{R}_{\mu \bar{\nu}} = \textrm{Ric}_{\mu \bar{\nu}}$.
So the problem reduces to taking the trace of $g_{\mu \bar{\nu}}^{\textrm{FS}}$.
My problem is that there are two ways to take the trace of the Ricci tensor that I can see, one raises the holomorphic index and then sums over antiholomorphic indices:
$$ R^{\bar{\sigma}}_{\; \;\bar{\nu}} = g^{\mu \bar{\sigma}} \textrm{Ric}_{\mu \bar{\nu}} $$
and the other raises the anti-holomorphic index and sums over the holomorphic indices:
$$ R^{\sigma}_{\; \;\mu} = g^{\sigma \bar{\nu}} \textrm{Ric}_{\mu \bar{\nu}} $$
I'm not sure which one is the correct expression to use to compute the Ricci scalar, any hints appreciated.
In this normalization the Ricci form is $(n + 1)$ times the Kähler form, i.e., $\rho = (n + 1)\omega$. Taking the trace with respect to $\omega$, the scalar curvature is $\sigma = (n + 1)n$ because the trace of the Kähler form with respect to itself is the complex dimension $n$.
Note carefully: The Fubini-Study metric is not geometrically unique, only unique up to scaling. There are different conventional choices depending on one's viewpoint. For some authors, the Kähler form is a generator of integral cohomology, i.e., $\rho = 2\pi(n + 1)\omega$; for others the holomorphic sectional curvature is equal to $4$; for others the metric is Einstein with Ricci form equal to the Kähler form; etc. All these metrics have the same Ricci form (representing $2\pi$ times the first Chern class of projective space), but the scalar curvature is inversely proportional to the overall scale of the metric.