Direct Image of Logarithmic Complexes and Infinitesimal Invariants of Cycles

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. 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.