The $L^2$ Volume of the Space of Holomorphic Maps from K\"ahler Riemann Surfaces to $\mathbb{CP}^k$

TL;DR

This paper proves a conjectural formula for the $L^2$ volume of the space of degree $r$ holomorphic maps from a compact K"ahler Riemann surface to complex projective space, confirming a previously posed conjecture.

Contribution

It provides a rigorous proof of the conjectural $L^2$ volume formula for holomorphic maps, extending prior special case verifications.

Findings

Confirmed the conjectural volume formula for general cases

Extended previous special case results

Strengthened understanding of the geometry of holomorphic mapping spaces

Abstract

We prove the conjectural formula for the $L^2$ volume of the space of degree $r$ holomorphic maps from a compact K\"ahler Riemann surface of genus $b$ to $\pk$. This formula was posed in \cite{Ba} and rigorously verified in \cite{Sp} for a special case using independent techniques.