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

Let $\mathcal{H}_{n,k}(Σ)$ be the space of degree $n\geq 1$ holomorphic maps from a compact Riemann surface $Σ$ to $\mathbb{C}P^k$. In the case $Σ=S^2$ 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 $k\geq 2$. 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.