Determinant line bundle on moduli space of parabolic bundles

TL;DR

This paper provides a direct construction of the Hermitian structure on the parabolic determinant line bundle, establishing a correspondence with Hermitian--Einstein connections using conical metrics and a parabolic Faltings' criterion.

Contribution

It introduces a direct method for constructing the Hermitian structure on the parabolic determinant line bundle, improving upon previous indirect approaches.

Findings

Established a direct Hermitian structure construction for parabolic determinant line bundles.

Extended the correspondence between stable parabolic bundles and Hermitian--Einstein connections to conical metrics.

Utilized a parabolic Faltings' criterion of semistability in the construction.

Abstract

In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian--Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings' criterion of semistability plays a crucial role.