Flat line bundles and the cappell-miller torsion in arakelov geometry

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-Soule 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 refine- ment 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/πi Z. In this context we prove an arithmetic Riemann-Roch theorem. This real- izes 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 Soule. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.