# Distances on the moduli space of complex projective structures

created by faraco on 09 Nov 2022

[BibTeX]

Published Paper

Inserted: 9 nov 2022
Last Updated: 9 nov 2022

Journal: Expositiones Mathematica
Year: 2019
Doi: https://doi.org/10.1016/j.exmath.2019.04.006

ArXiv: 1812.00695 PDF

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.

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