A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers

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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 [5]. 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.