Direct image of logarithmic complexes and infinitesimal invariants of cycles

TL;DR

This paper demonstrates how the direct image of filtered logarithmic de Rham complexes decomposes into simpler components, enabling the study of infinitesimal invariants of algebraic cycles through Hodge theory.

Contribution

It generalizes the decomposition theorem to filtered D-modules and shows how logarithmic complexes can be used to analyze infinitesimal invariants of cycles.

Findings

Decomposition of direct images into sums of filtered complexes.

Strictness of Hodge filtration after cohomology in the projective case.

Equivalence of infinitesimal invariants to cohomology classes of cycles.

Abstract

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.