  Let $f: X \to S$ be a smooth morphism in characteristic 0, and let $(E,
\nabla_{X/S})$ be a relative regular connection. We define a cohomology of
relative differential characters on $X$ which receives classes of $(E,
\nabla_{X/S})$. It says in particular that the partial vanishing of the trace
of the iterated Atiyah classs can be made canonical. While applied to a family
of curves and $c_2$, the construction yields a connection of $f_*c_2(E)$. This
one has been constructed analytically by A. Beilinson. We also relate our
construction to the trace complex of Beilinson of Schechtman.
