The Prym vector bundle and Gunning vector bundle over the Teichmüller space

We introduce and study the Prym vector bundle $P$ of holomorphic Prym differentials and the Ganning cohomology bundle $G$ over the Teichmueller space of compact Riemann surfaces of genus $g\geq 2$ and over the Torelli space of genus $g\geq 2$. We construct a basis of holomorphic Prym differentials on a variable compact Riemann surface which depends on the moduli of the compact Riemann surface and on the essential characters. From these bundles we compose an exact sequence of holomorphic vector bundles over the product of the Teichmueller space of genus $g$ and a special domain in the complex manifold $\mathbb C^{2g}/\mathbb Z{2g}$.