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

TL;DR

This paper extends Arakelov geometry to flat line bundles, establishing a new arithmetic Riemann-Roch theorem and linking holomorphic torsion with intersection theory and periods of differential forms.

Contribution

It introduces a novel Arakelov theory for flat line bundles, connecting Cappell-Miller torsion with intersection connections and arithmetic intersection numbers.

Findings

Proves an arithmetic Riemann-Roch theorem for flat line bundles.

Establishes equivalence between Cappell-Miller torsion and holomorphic determinants.

Provides examples illustrating invariants like periods of differential forms.

Abstract

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 star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology side, the idea goes back to Fay's holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in ${\mathbb C}/\pi i {\mathbb Z}$. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that the holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soule. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.