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

TL;DR

This paper explicitly computes the Ricci and scalar curvatures, as well as the Einstein-Hilbert action, for the space of degree one holomorphic maps from S^2 to CP^k, confirming a conjectured formula.

Contribution

It provides the first explicit calculations of the Einstein-Hilbert action for this space, extending previous work on the L^2 metric to curvature and action computations.

Findings

Ricci curvature tensor and scalar curvature are explicitly determined for the space.

The Einstein-Hilbert action is computed exactly and matches the conjectured formula.

Results extend understanding of geometric structures on holomorphic map spaces.

Abstract

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