Asian Journal of Mathematics

Volume 25 (2021)

Number 4

Meromorphic connections, determinant line bundles and the Tyurin parametrization

Pages: 455 – 476

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n4.a1

Authors

Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Jacques Hurtubise (Department of Mathematics, McGill University, Montreal, Quebec, Canada)

Abstract

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.

Keywords

Tyurin parametrization, meromorphic connection, framing, stable bundle, symplectic form, Atiyah bundle

2010 Mathematics Subject Classification

14D20, 34M40, 53D30

Received 1 June 2019

Accepted 14 October 2020

Published 25 April 2022