The space of harmonic Prym differentials on a variable compact Riemann surface

The harmonic Prym differentials and their period classes play an important role in the modern theory of functions on compact Riemann surfaces [1-7]. We study the harmonic Prym bundle, whose fibers are the spaces of harmonic Prym differentials on variable compact Riemann surfaces and find its connection with Gunning's cohomological bundle over the Teichmuller space for two important subgroups of the inessential and normalized characters on a compact Riemann surface. We study the periods of holomorphic Prym differentials for essential characters on variable compact Riemann surfaces.