Title: Local Index Theory and Higher Analytic Torsion

In this paper, 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 Soul\'e 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.

1991 Mathematics Subject Classification: 32L10,57R20,58G10

Keywords and Phrases: Sheaves and cohomology of sections of holomorphic vector bundles. Characteristic classes and numbers. Index theory and related fixed point theory.

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