  We develop a holomorphic equivalence between on one hand the space of pairs
(stable bundle, flat connection on the bundle) and the "sheaf of holomorphic
connections" (the sheaf of splittings of the one-jet sequence) for the
determinant (Quillen) line bundle over the moduli space of vector bundles on a
compact connected Riemann surface. This equivalence is shown to be
holomorphically symplectic. The equivalences, both holomorphic and symplectic,
seem to be quite general, in that they extend to other general families of
holomorphic bundles and holomorphic connections, in particular those arising
from "Tyurin families" of stable bundles over the surface. These families
generalize the Tyurin parametrization of stable vector bundles $E$ over a
compact connected Riemann surface, and one can build above them spaces of
(equivalence classes of) connections, which are again symplectic. These spaces
are also symplectically biholomorphically equivalent to the sheaf of
connections for the determinant bundle over the Tyurin family. The last portion
of the paper shows how this extends to moduli of framed bundles.
