A Hodge module construction of relative Du Bois complexes

RESUMO

We construct relative Du Bois and relative intersection Du Bois complexes for smooth-factorizable morphisms to a smooth complex base using relative de Rham complexes of mixed Hodge modules and Grothendieck duality. For a smooth morphism, these complexes recover the relative Kähler differentials. Comparisons with the relative Du Bois complexes of Kovács and Ning remain open. We construct finite filtrations on the absolute Du Bois and intersection Du Bois complexes whose graded pieces are the corresponding relative complexes tensored with differential forms from the base. We also study restriction to fibers and its relation to symbolic $m$-Du Bois and weak $m$-rational singularities.