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 ( M , F ∙ ) {\displaystyle (M,F^{\bullet })} together with a perverse sheaf F {\displaystyle {\mathcal {F}}} such that the functor from the Riemann–Hilbert correspondence sends ( M , F ∙ ) {\displaystyle (M,F^{\bullet })} to F {\displaystyle {\mathcal {F}}} . 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 X {\displaystyle X} there is an abelian category MHM ( X ) {\displaystyle {\textbf {MHM}}(X)} pg 339 with the following functorial properties
There is a faithful functor rat X : MHM ( X ) → Perv ( X ; Q ) {\displaystyle {\text{rat}}_{X}:{\textbf {MHM}}(X)\to {\textbf {Perv}}(X;\mathbb {Q} )} called the rationalization functor. This gives the underlying rational perverse sheaf of a mixed Hodge module. There is a faithful functor Dmod X : MHM ( X ) → Mod ( D X ) {\displaystyle {\text{Dmod}}_{X}:{\textbf {MHM}}(X)\to {\textbf {Mod}}({\mathcal {D}}_{X})} sending a mixed Hodge module to its underlying D-module. These functors behave well with respect to the Riemann-Hilbert correspondence D R X : D C o h b ( D X ) → D c s b ( X ; C ) {\displaystyle DR_{X}:D_{Coh}^{b}({\mathcal {D}}_{X})\to D_{cs}^{b}(X;\mathbb {C} )} , meaning for every mixed Hodge module M {\displaystyle M} there is an isomorphism α : rat X ( M ) ⊗ C → ∼ DR X ( Dmod X ( M ) ) {\displaystyle \alpha :{\text{rat}}_{X}(M)\otimes \mathbb {C} \xrightarrow {\sim } {\text{DR}}_{X}({\text{Dmod}}_{X}(M))} . In addition, there are the following categorical properties
The category of mixed Hodge modules over a point is isomorphic to the category of Mixed hodge structures, MHM ( { p t } ) ≅ MHS {\displaystyle {\textbf {MHM}}(\{pt\})\cong {\text{MHS}}}
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