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

Meromorphic connections, determinant line bundles and the Tyurin parametrization