Periods of abelian differentials with prescribed singularities

Abstract

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 multiplicites of zeros. Our main result was also obtained, independently, by Bainbridge, Johnson, Judge and Park.