Mixed Hodge module

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 formula_1 together with a perverse sheaf formula_2 such that the functor from the Riemann–Hilbert correspondence sends formula_1 to formula_2. 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. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves.
Abstract structure.
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 formula_5 there is an abelian category formula_6pg 339 with the following functorial properties
In addition, there are the following categorical properties
For a morphism formula_21 of algebraic varieties, the associated six functors on formula_22 and formula_23 have the following properties
Relation between derived categories.
The derived category of mixed Hodge modules formula_22 is intimately related to the derived category of constructuctible sheaves formula_29 equivalent to the derived category of perverse sheaves. This is because of how the rationalization functor is compatible with the cohomology functor formula_30 of a complex formula_25 of mixed Hodge modules. When taking the rationalization, there is an isomorphismformula_32for the middle perversity formula_33. Notepg 310 this is the function formula_34 sending formula_35, which differs from the case of pseudomanifolds where the perversity is a function formula_36 where formula_37. Recall this is defined as taking the composition of perverse truncations with the shift functor, sopg 341formula_38This kind of setup is also reflected in the derived push and pull functors formula_39 and with nearby and vanishing cycles formula_40, the rationalization functor takes these to their analogous perverse functors on the derived category of perverse sheaves.
Tate modules and cohomology.
Here we denote the canonical projection to a point by formula_41. One of the first mixed Hodge modules available is the weight 0 Tate object, denoted formula_42 which is defined as the pullback of its corresponding object in formula_43, soformula_44It has weight zero, so formula_45 corresponds to the weight 0 Tate object formula_46 in the category of mixed Hodge structures. This object is useful because it can be used to compute the various cohomologies of formula_5 through the six functor formalism and give them a mixed Hodge structure. These can be summarized with the tableformula_48Moreover, given a closed embedding formula_49 there is the local cohomology groupformula_50
Variations of Mixed Hodge structures.
For a morphism of varieties formula_51 the pushforward maps formula_52 and formula_53 give degenerating variations of mixed Hodge structures on formula_54. In order to better understand these variations, the decomposition theorem and intersection cohomology are required.
Intersection cohomology.
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 formula_51 of varieties. To define the intersection complex, let formula_56 be the open smooth part of a variety formula_5. Then the intersection complex of formula_5 can be defined asformula_59whereformula_60as with perverse sheavespg 311. In particular, this setup can be used to show the intersection cohomology groupsformula_61have a pure weight formula_62 Hodge structure.