The Einstein-Hilbert action of the space of holomorphic maps from S^2 to CP^k

Let ${\mathcal{H}}_{n,k}(\mathrm{\Sigma })$ be the space of degree $n\ge 1$ holomorphic maps from a compact Riemann surface $\mathrm{\Sigma }$ to $\mathbb{C}{P}^k$. In the case $\mathrm{\Sigma }=S^2$ and $n=1$, the $L^2$ metric on ${\mathcal{H}}_{1,k}(S^2)$ was computed exactly by Speight. In this paper, the Ricci curvature tensor and the scalar curvature on ${\mathcal{H}}_{1,k}(S^2)$ are determined explicitly for $k\ge 2$. An exact direct computation of the Einstein-Hilbert action with respect to the $L^2$ metric on ${\mathcal{H}}_{1,k}(S^2)$ is made and shown to coincide with a formula conjectured by Baptista.