The Einstein-Hilbert action of the space of holomorphic maps from S2 to CPk

Abstract

Let Hn,k(σ) be the space of degree n ≥ 1 holomorphic maps from a compact Riemann surface σ to CPk. In the case σ = S2 and n = 1, the L2 metric on H1,k(S2) was computed exactly by Speight. In this paper, the Ricci curvature tensor and the scalar curvature on H1,k(S2) are determined explicitly for k ≥ 2. An exact direct computation of the Einstein-Hilbert action with respect to the L2 metric on H1,k(S2) is made and shown to coincide with a formula conjectured by Baptista. © 2013 Elsevier B.V.