The Einstein–Hilbert action of the space of holomorphic maps from S2S2 to CPkCPk

Let Hn,k(Σ)Hn,k(Σ) be the space of degree n≥1n≥1 holomorphic maps from a compact Riemann surface ΣΣ to CPkCPk. In the case Σ=S2Σ=S2 and n=1n=1, the L2L2 metric on H1,k(S2)H1,k(S2) was computed exactly by Speight. In this paper, the Ricci curvature tensor and the scalar curvature on H1,k(S2)H1,k(S2) are determined explicitly for k≥2k≥2. An exact direct computation of the Einstein–Hilbert action with respect to the L2L2 metric on H1,k(S2)H1,k(S2) is made and shown to coincide with a formula conjectured by Baptista.