  We introduce the notion of mixed Hodge complex on an algebraic variety,
improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge
structure with the theory of mixed Hodge module. This was supposed to be true,
but is quite nontrivial, because the theory of mixed Hodge module does not work
on a simplicial scheme. Some application to the Du Bois singularity is given.
