We give a necessary and sufficient condition for a representation of the fundamental group of a closed surface of genus at least $2$ to ${\mathbb{C}}$ to be the holonomy of a translation surface with a prescribed list of conical singularities. Equivalently, we determine the period maps of abelian differentials with prescribed list of multiplicities of zeros. Our main result was also obtained, independently, by Bainbridge, Johnson, Judge, and Park.

We characterize the representations of the fundamental group of a closed surface to PSL 2 (C) that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces.

We define a class of representations of the fundamental group of a closed surface of genus 2 to PSL 2 (C): the pentagon representations. We show that they are exactly the non-elementary PSL 2 (C)-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to PSL 2 (C) is the holonomy of a projective structure, possibly with one branched point of order 2. We show that pentagon representations arise as such holonomies and repair their proof.Résumé. -Nous définissons une classe de représentations du groupe fondamental d'une surface fermée de genre 2 dans PSL 2 (C) : les représentations pentagones. Nous montrons que ce sont exactement les représentations d'un groupe de surface dans PSL 2 (C) qui n'admettent pas de décomposition de Schottky, i.e. de décomposition en pantalons telle que la restriction de la représentation à chaque pantalon est un isomorphisme sur un groupe de Schottky. Ce faisant, nous exhibons une lacune dans la preuve de Gallo, Kapovich et Marden du fait que toute représentation non-élémentaire d'un groupe de surface dans PSL 2 (C) est l'holonomie d'une structure projective, avec éventuellement un point de branchement d'ordre 2. Nous montrons que les représentations pentagones sont de telles holonomies et réparons leur preuve.

We characterize the representations of the fundamental group of a closed surface to PSL 2 (ℂ) that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces.