  We show that the direct image of the filtered logarithmic de Rham complex is
a direct sum of filtered logarithmic complexes with coefficients in variations
of Hodge structures, using a generalization of the decomposition theorem of
Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The
advantage of using the logarithmic complexes is that we have the strictness of
the Hodge filtration by Deligne after taking the cohomology group in the
projective case. As a corollary, we get the total infinitesimal invariant of a
(higher) cycle in a direct sum of the cohomology of filtered logarithmic
complexes with coefficients, and this is essentially equivalent to the
cohomology class of the cycle.
