Abstract
For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. 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.
Funding Statement
G. F. was supported in part by ANR grant ANR-12-BS01-0002. R. W. was supported in part by NSF grant DMS-1406513. The authors also acknowledge support from NSF grants DMS-1107452, DMS-1107263, and DMS-1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
Citation
Gerard Freixas i Montplet. Richard A. Wentworth. "Deligne pairings and families of rank one local systems on algebraic curves." J. Differential Geom. 115 (3) 475 - 528, July 2020. https://doi.org/10.4310/jdg/1594260017