Period-preserving variation of a Riemann surface (Q1321781)

In the Teichmüller space of a given surface, the sub-locus, consisting of all marked Riemann surfaces which admit holomorphic abelian differentials having the prescribed periods, plays several important roles in the theory of Riemann surfaces. In this note, we will introduce explicit local parameters at a generic point of this sub-locus. They indicate how zeros of differentials with prescribed periods vary on the surfaces.