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 $\mathcal{P}\left(S\right)$ of marked isomorphism classes of projective structures on $S$. We show that $\mathcal{P}\left(S\right)$, endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichmüller space $\mathcal{T}\left(S\right)$ of $S$. We shall notice also that the Kobayashi and Carathéodory pseudodistances, which can be defined for any complex manifold, cannot be upgraded to a distance. We finally show that $\mathcal{P}\left(S\right)$ does not carry any Bergman pseudometric.

中文翻译：

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复杂射影结构模空间上的距离

让 $\mathrm{小号}$ 是属的闭合且定向的表面 $G$ 至少2。在这篇（主要是说明性的）文章中，研究的对象是空间 $\mathcal{P}（\mathrm{小号}）$ 上射影结构的显着同构类 $\mathrm{小号}$。我们证明$\mathcal{P}（\mathrm{小号}）$具有规范的复杂结构，带有奇异的埃尔米特式结构，在Teichmüller空间扩展了经典结构 $\mathcal{Ť}（\mathrm{小号}）$ 的 $\mathrm{小号}$。我们还将注意到，可以为任何复杂流形定义的Kobayashi和Carathéodory伪距不能升级到一定距离。我们终于证明了$\mathcal{P}（\mathrm{小号}）$ 不带有任何Bergman伪度量。