# The direct image of a flat fibration with complex fibers

## TL;DR

This paper develops a method to construct odd characteristic classes for proper flat fibrations with complex fibers and provides a Riemann-Roch-Grothendieck theorem for calculating these classes of flat vector bundles derived from fiberwise holomorphic bundles.

## Contribution

It introduces a generalized construction of odd characteristic classes for such fibrations and establishes a Riemann-Roch-Grothendieck theorem for their computation.

## Key findings

- Constructed odd characteristic classes for flat fibrations with complex fibers.
- Established a Riemann-Roch-Grothendieck theorem for flat vector bundles.
- Extended Bismut-Lott methods to new classes of fibrations.

## Abstract

We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut-Lott. Then we consider the direct image of a fiberwise holomorphic vector bundle, which is a flat vector bundle on the base. We give a Riemann-Roch-Grothendieck theorem calculating the odd real characteristic classes of this flat vector bundle.