Deligne pairings and families of rank one local systems on algebraic curves

TL;DR

This paper extends intersection theory to line bundles with flat connections on algebraic curve families, establishing a canonical connection on the Deligne pairing and exploring its relations with analytic torsion and hyperholomorphic line bundles.

Contribution

It introduces a functorial, canonical 'intersection' connection on Deligne pairings for flat line bundles, linking it to analytic torsion and hyperholomorphic geometry.

Findings

Existence of a canonical, functorial intersection connection on Deligne pairings.

Relation between the intersection connection and the holomorphic extension of analytic torsion.

Construction of a meromorphic connection on hyperholomorphic line bundles over twistor spaces.

Abstract

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.