Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

Freixas i Montplet, Gerard; Wentworth, Richard A.

doi:10.24033/asens.2409

Flat line bundles and the Cappell-Miller torsion in Arakelov geometry
[Fibrés en droites plats et torsion de Cappell-Miller en géométrie d'Arakelov]

Freixas i Montplet, Gerard ; Wentworth, Richard A.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1265-1303.

Résumé
Abstract

Dans cet article nous étendons l'isomorphisme de Riemann-Roch fonctoriel pour les fibrés en droites holomorphes Hermitiens, dû à Deligne, au cas des fibrés plats non nécessairement unitaires. La métrique de Quillen et le produit $☆$ de Gillet-Soulé sont remplacés par des logarithmes à valeurs complexes. Sur le déterminant de la cohomologie, nous montrons que la torsion de Cappell-Miller est l'analogue approprié de la métrique de Quillen. Sur les accouplements de Deligne, les logarithmes raffinent les connexions d'intersection introduites dans un travail précédent. La construction conduit naturellement à une théorie d'Arakelov pour les fibrés plats sur les surfaces arithmétiques, et produit des nombres d'intersection arithmétique à valeurs dans $ℂ / π i ℤ$ . Dans ce contexte, nous démontrons une formule de Riemann-Roch arithmétique. On réalise ainsi un programme proposé par Cappell-Miller visant à montrer que leur torsion holomorphe possède des propriétés analogues à celles de la métrique de Quillen établies par Bismut, Gillet et Soulé. Finalement, nous donnons des exemples qui clarifient le type d'invariants que ce formalisme encode: des périodes de formes différentielles.

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for Hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and $☆$ -product of Gillet-Soulé are replaced with complex valued logarithms. On the determinant of cohomology side, we show that the Cappell-Miller torsion is the appropriate counterpart of the Quillen metric. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in a previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in $ℂ / π i ℤ$ . In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that their holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soulé. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.

Publié le : 2019-11-12

MR Zbl

DOI : 10.24033/asens.2409

Classification : 58J52; 14C40.
Keywords: Flat connections, Cappell-Miller torsion, Riemann-Roch, arithmetic intersections.
Mot clés : Connexions plates, torsion de Cappell-Miller, Riemann-Roch, intersections arithmétiques.

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     title = {Flat line bundles and the {Cappell-Miller} torsion in {Arakelov} geometry},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1265--1303},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Freixas i Montplet, Gerard; Wentworth, Richard A. Flat line bundles and the Cappell-Miller torsion in Arakelov geometry. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1265-1303. doi : 10.24033/asens.2409. https://www.numdam.org/articles/10.24033/asens.2409/

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