An arithmetic Lefschetz-Riemann-Roch theorem. With an appendix by Xiaonan Ma

TL;DR

This paper establishes an arithmetic Lefschetz-Riemann-Roch theorem within Arakelov geometry, extending fixed point formulas to higher equivariant arithmetic K-theory for regular projective arithmetic schemes with group actions.

Contribution

It introduces a new higher equivariant arithmetic K-theory Lefschetz-Riemann-Roch theorem generalizing previous fixed point formulas.

Findings

Defines a direct image map for equivariant morphisms in higher arithmetic K-groups.

Proves transitivity property of the direct image map.

Establishes the arithmetic Lefschetz-Riemann-Roch theorem using localization and concentration techniques.

Abstract

In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic concentration theorem developed in \cite{T3} to prove an arithmetic Lefschetz-Riemann-Roch theorem. This theorem can be viewed as a generalization, to the higher equivariant arithmetic K-theory, of the fixed point formula of Lefschetz type proved by K. K\"{o}hler and D. Roessler in \cite{KR1}.