Is there any easy way to compute the determinant of the Fubini-Study metric, given by:
$g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha z_{\bar{\beta}}}{1+\bar{z}z}\right)$ ?
I've tried using the relation of the determinant and the Levi-Civita connections $\Gamma^\gamma_{\bar{\beta}\gamma}$ (since my goal at the end would be to compute the Ricci curvature tensor $R_{\alpha\bar{\beta}}$). The Ricci curvature tensor is given by:
$R_{\alpha\bar{\beta}}=-\partial_\alpha\partial_{\bar{\beta}}\left(\log(\det g_{\gamma\bar{\delta}})\right)=-\partial_\alpha\left(g^{\gamma\bar{\delta}}\partial_\bar{\beta}g_{\gamma\bar{\delta}}\right).$
Where I got the last equality by using the Levi-Civita connection do solve the determinant. Now I'm still having problems with the calculations of the last term, any hints on that one ?
Any help would be most welcome!
A solution to the determinant would be most helpfull since I think I could compare this with the Kähler-potential given by $K(z,\bar{z})=\log(1+z\bar{z})$, for which we have that $g_{\alpha\bar{\beta}}=\partial_\alpha\partial_{\bar{\beta}}K(z,\bar{z})$.
Edit, since I've seen people asking this question before. I'm working on a complex manifold with coordinates $z^a=\{z^\alpha,\bar{z}^\bar{\alpha}\}$, where $a=1,...,2n$ and $\alpha=1,...,n$. So when I write Greek indices, I've split up the 2n linear independent (denoted by an $a$) into two groups, the holomorphic (denoted by $\alpha$) and anti-holomorphic (denoted by $\bar{\alpha}$) coordinates.