Abstract
For any prescribed differential principal part on a compact Riemann surface, there are uniquely determined and intrinsically defined meromorphic abelian differentials with these principal parts, defined independently of any choice of a marking of the surface or of a basis for the holomorphic abelian differentials, and they are holomorphic functions of the singularities. They can be constructed explicitly in terms of intrinsically defined cross-ratio functions on the Riemann surfaces, the classical cross-ratio function for the Riemann sphere, and natural generalizations for surfaces of higher genus.
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