Kai Köhler

Abstract:   The complex Ray-Singer analytic torsion is a linear combination of regularized determinants of Laplace operators. Its main application is related to the Arakelov geometry of flat schemes over Spec Z, equipped with Hermitian metrics at the manifolds of their complex points. For proper morphisms between such arithmetic varieties, the analytic torsion provides a push forward of Hermitian vector bundles over the varieties. In the case of a group scheme action on the arithmetic varieties, it turns out that this push forward map localizes on the fixed point scheme. This can be seen as an analogue both in differential geometry and arithmetic algebraic geometry of the classical fixed point theorem for the Lefschetz number.

One obtains several statements about analytic torsion as applications of this result. The arithmetical applications include a new proof of the Jantzen sum formula for lattice representations of Chevalley schemes, a topological formula for global heights of all generalized flag spaces, and a new proof for Colmez' formula for the Faltings heights of abelian varieties with complex multiplication. Another application is a Bott residue formula for Gillet-Soulé's arithmetic characteristic classes.