We report on the construction of secondary invariants in connection with the Atiyah-Singer index theorem for families, and the Theorem of Riemann-Roch-Grothendieck. The local families index theorem plays an important role in the construction. In complex geometry, the corresponding objects are the analytic torsion forms and the analytic torsion currents. These objects exhibit natural functorial properties with respect to composition of maps. Gillet and Soulé have used these objects to prove a Riemann-Roch theorem in Arakelov geometry. Also we state a Riemann-Roch theorem for flat vector bundles, and report on the construction of corresponding higher analytic torsion forms.

58J20, 58J52, 32L10

sheaves and cohomology of sections of holomorphic vector bundles, characteristic classes and numbers, secondary invariant, Atiyah-Singer index theorem, theorem of Riemann-Roch-Grothendieck, local families index theorem, analytic torsion forms, analytic torsion currents, flat vector bundles