# Distances on the moduli space of complex projective structures

## TL;DR

This paper explores the complex structure and metric properties of the space of projective structures on a surface, revealing exotic hermitian structures and limitations of certain pseudodistances.

## Contribution

It introduces exotic hermitian structures on the moduli space of projective structures and analyzes the non-existence of certain pseudometrics like the Bergman metric.

## Key findings

- Exotic hermitian structures extend classical ones on Teichmüller space.
- Kobayashi and Carathéodory pseudodistances cannot be upgraded to true distances.
- The space does not admit a Bergman pseudometric.

## Abstract

Let $S$ be a closed and oriented surface of genus $g$ at least $2$. In this (mostly expository) article, the object of study is the space $\mathcal{P}(S)$ of marked isomorphism classes of projective structures on $S$. We show that $\mathcal{P}(S)$, endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichm\"uller space $\mathcal{T}(S)$ of $S$. We shall notice also that the Kobayashi and Carath\'eodory pseudodistances, which can be defined for any complex manifold, can not be upgraded to a distance. We finally show that $\mathcal{P}(S)$ does not carry any Bergman pseudometric.