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 P(S) of marked isomorphism classes of projective structures on S. We show that P(S), endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichmuller space T(S) of S. We shall notice also that the Kobayashi and Caratheodory pseudodistances, which can be defined for any complex manifold, cannot be upgraded to a distance. We finally show that P(S) does not carry any Bergman pseudometric. (C) 2019 Elsevier GmbH. All rights reserved.