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July 2020 Deligne pairings and families of rank one local systems on algebraic curves
Gerard Freixas i Montplet, Richard A. Wentworth
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J. Differential Geom. 115(3): 475-528 (July 2020). DOI: 10.4310/jdg/1594260017


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).


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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.


Received: 25 January 2017; Published: July 2020
First available in Project Euclid: 9 July 2020

zbMATH: 07225029
MathSciNet: MR4120817
Digital Object Identifier: 10.4310/jdg/1594260017

Primary: 58J52
Secondary: 14C40

Rights: Copyright © 2020 Lehigh University


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Vol.115 • No. 3 • July 2020
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