Wikipedia / Mixed Hodge module
Wikipedia / 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 together with a perverse sheaf such that the functor from the Riemann–Hilbert correspondence. 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 varietypg 339 with the following functorial properties
There is a faithful functor called the rationalization functor. This gives the underlying rational perverse sheaf of a mixed Hodge module.
sending a mixed Hodge module to its underlying D-module.
.
In addition, there are the following categorical properties
admits a weight filtration such that every morphism are semi-simple, and in the category of mixed Hodge modules over a point, this corresponds to the weight filtration of a mixed Hodge structure.
There is a dualizing functor.
have the following properties
of mixed Hodge modules.
of mixed Hodge modules.
Relation between derived categories
The derived category of mixed Hodge modules. When taking the rationalization, there is an isomorphism. Notepg 310, which differs from the case of pseudomanifolds. Recall this is defined as taking the composition of perverse truncations with the shift functor, sopg 341, the rationalization functor takes these to their analogous perverse functors on the derived category of perverse sheaves.
Tate modules and cohomology
, so through the six functor formalism and give them a mixed Hodge structure. These can be summarized with the table there is the local cohomology group
Variations of Mixed Hodge structures
. In order to better understand these variations, the decomposition theorem and intersection cohomology are required.
Intersection cohomology
can be defined aswhereas with perverse sheavespg 311. In particular, this setup can be used to show the intersection cohomology groups Hodge structure.
See also
References