Distances on the moduli space of complex projective structures

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.