In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism.
Essentially, these objects are a pair of a filtered D-module
such that the functor from the Riemann–Hilbert correspondence sends
This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered.
This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure.
[1] This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves.
Before going into the nitty gritty details of defining mixed Hodge modules, which is quite elaborate, it is useful to get a sense of what the category of mixed Hodge modules actually provides.
Given a complex algebraic variety
The derived category of mixed Hodge modules
is intimately related to the derived category of constructible sheaves
equivalent to the derived category of perverse sheaves.
When taking the rationalization, there is an isomorphism
, which differs from the case of pseudomanifolds where the perversity is a function
Recall this is defined as taking the composition of perverse truncations with the shift functor, so[2]pg 341
This kind of setup is also reflected in the derived push and pull functors
, the rationalization functor takes these to their analogous perverse functors on the derived category of perverse sheaves.
Here we denote the canonical projection to a point by
One of the first mixed Hodge modules available is the weight 0 Tate object, denoted
which is defined as the pullback of its corresponding object in
corresponds to the weight 0 Tate object
in the category of mixed Hodge structures.
This object is useful because it can be used to compute the various cohomologies of
through the six functor formalism and give them a mixed Hodge structure.
there is the local cohomology group
give degenerating variations of mixed Hodge structures on
In order to better understand these variations, the decomposition theorem and intersection cohomology are required.
One of the defining features of the category of mixed Hodge modules is the fact intersection cohomology can be phrased in its language.
This makes it possible to use the decomposition theorem for maps
be the open smooth part of a variety
In particular, this setup can be used to show the intersection cohomology groups