  We prove the Bloch conjecture : $ c_2(E) \in H^4_\cald (X,\bbz(2))$ is
torsion for holomorphic rank two vector bundles $E$ with an integrable
connection over a complex projective variety $X$. We prove also the rationality
of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds.
We give a sharp higher-dimensional Milnor inequality for the volume regulator
of all representations to $PSO(1,n)$ of fundamental groups of compact
$n$-dimensional hyperbolic manifolds, announced in our earlier paper.
