Asian Journal of Mathematics
Volume 25 (2021)
Meromorphic connections, determinant line bundles and the Tyurin parametrization
Pages: 455 – 476
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 holomorphic 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, are rather quite general, for example, 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) holomorphic connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of holomorphic connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.
Tyurin parametrization, meromorphic connection, framing, stable bundle, symplectic form, Atiyah bundle
2010 Mathematics Subject Classification
14D20, 34M40, 53D30
Indranil Biswas is partially supported by a J. C. Bose Fellowship.
Received 1 June 2019
Accepted 14 October 2020
Published 25 April 2022