Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles

TL;DR

This paper proves the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles using local index theory, eta forms, and Cheeger-Chern-Simons classes in even-dimensional fibers.

Contribution

It provides a differential form level proof of the theorem for complex flat vector bundles without kernel bundle assumptions in even-dimensional fibers.

Findings

Proof of the real part of the Riemann-Roch-Grothendieck theorem in this setting

Application of local family index theorem to perturbed twisted spin Dirac operators

Development of a variational formula for the Bismut-Cheeger eta form

Abstract

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut-Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger-Chern-Simons class of complex flat vector bundle.