Meromorphic connections, determinant line bundles and the Tyurin parametrization

Abstract

We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the of holomorphic (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 of holomorphic bundles and holomorphic connections, in particular those arising from families of stable bundles over the surface. These 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.