Complex Algebraic Geometry

Characteristic classes of Flat bundles on smooth algebraic varieties are defined in various cohomology theories. We consider the de Rham cohomology, the Deligne cohomology and the rational Chow groups and study the classes. We focus on the special flat bundles which arise from smooth families, namely the cohomology bundles or the de Rham bundles. We consider the parabolic canonical extensions of these bundles on a nonsingular compactification of the base and obtain a relation between the parabolic Chern character of the different weights de Rham bundles in the rational Chow groups. This is a joint work with C. Simpson.