A Riemann-Roch Theorem for Flat Bundles, with Values in the Algebraic Chern-Simons Theory

Abstract

Our purpose in this paper is to continue the algebraic study of complex local systems on complex algebraic varieties. We prove a Riemann-Roch theorem for these objects using algebraic Chern-Simons characteristic classes. A complex local system E on a smooth, projective complex variety X gives rise to a locally free analytic sheaf Ean := E ⊗C Oan X which (using GAGA) admits a canonical algebraic structure E. The tautological analytic connection on E⊗CO X induces an integrable algebraic connection∇ : E → E⊗ΩX . Combining GAGA with the Poincare lemma, we see that the analytic cohomology of the local system can be identified with the hypercohomology of the algebraic de Rham complex