Mixed Hodge module

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 ${\displaystyle X}$ there is an abelian category ${\displaystyle {\textbf {MHM}}(X)}$^{[2]pg 339} with the following functorial properties

1. There is a faithful functor ${\displaystyle {\text{rat}}_{X}:{\textbf {MHM}}(X)\to {\textbf {Perv}}(X;\mathbb {Q} )}$ called the rationalization functor. This gives the underlying rational perverse sheaf of a mixed Hodge module.
2. There is a faithful functor ${\displaystyle {\text{Dmod}}_{X}:{\textbf {MHM}}(X)\to {\textbf {Mod}}({\mathcal {D}}_{X})}$ sending a mixed Hodge module to its underlying D-module.
3. These functors behave well with respect to the Riemann-Hilbert correspondence ${\displaystyle DR_{X}:D_{Coh}^{b}({\mathcal {D}}_{X})\to D_{cs}^{b}(X;\mathbb {C} )}$, meaning for every mixed Hodge module ${\displaystyle M}$ there is an isomorphism ${\displaystyle \alpha$ :{\text{rat}}_{X}(M)\otimes \mathbb {C} \xrightarrow {\sim } {\text{DR}}_{X}({\text{Dmod}}_{X}(M))} .

In addition, there are the following categorical properties

1. The category of mixed Hodge modules over a point is isomorphic to the category of Mixed hodge structures, ${\displaystyle {\textbf {MHM}}(\{pt\})\cong {\text{MHS}}}$
2. Every object ${\displaystyle M}$ in ${\displaystyle {\textbf {MHM}}(X)}$ admits a weight filtration ${\displaystyle W}$ such that every morphism in ${\displaystyle {\textbf {MHM}}(X)}$ preserves the weight filtration strictly, the associated graded objects ${\displaystyle {\text{Gr}}_{k}^{W}(M)}$ 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.
3. There is a dualizing functor ${\displaystyle \mathbb {D} _{X}}$ lifting the Verdier dualizing functor in ${\displaystyle D_{cs}^{b}(X;\mathbb {Q} )}$ which is an involution on ${\displaystyle {\textbf {MHM}}(X)}$.

For a morphism ${\displaystyle f:X\to Y}$ of algebraic varieties, the associated six functors on ${\displaystyle D^{b}{\textbf {MHM}}(X)}$ and ${\displaystyle D^{b}{\textbf {MHM}}(Y)}$ have the following properties

1. ${\displaystyle f_{!},f^{*}}$ don't increase the weights of a complex ${\displaystyle M^{\bullet }}$ of mixed Hodge modules.
2. ${\displaystyle f^{!},f_{*}}$ don't decrease the weights of a complex ${\displaystyle M^{\bullet }}$ of mixed Hodge modules.

The derived category of mixed Hodge modules ${\displaystyle D^{b}{\textbf {MHM}}(X)}$ is intimately related to the derived category of constructible sheaves ${\displaystyle D_{cs}^{b}(X;\mathbb {Q} )\cong D^{b}({\text{Perv}}(X;\mathbb {Q} ))}$ equivalent to the derived category of perverse sheaves. This is because of how the rationalization functor is compatible with the cohomology functor ${\displaystyle H^{k}}$ of a complex ${\displaystyle M^{\bullet }}$ of mixed Hodge modules. When taking the rationalization, there is an isomorphism

${\displaystyle {\text{rat}}_{X}(H^{k}(M^{\bullet }))={\text{ }}^{\mathbf {p} }H^{k}({\text{rat}}_{X}(M^{\bullet }))}$

for the middle perversity ${\displaystyle \mathbb {p} }$. Note^{[2]pg 310} this is the function ${\displaystyle \mathbf {p} :2\mathbb {N} \to \mathbb {Z} }$ sending ${\displaystyle \mathbf {p} (2k)=-k}$, which differs from the case of pseudomanifolds where the perversity is a function ${\displaystyle \mathbb {p}$ :[2,n]\to \mathbb {Z} _{\geq 0}} where ${\displaystyle \mathbf {p} (2k)=\mathbf {p} (2k-1)=k-1}$. Recall this is defined as taking the composition of perverse truncations with the shift functor, so^{[2]pg 341}

${\displaystyle {\text{ }}^{\mathbf {p} }H^{k}({\text{rat}}_{X}(M^{\bullet }))={\text{ }}^{\mathbf {p} }\tau _{\leq 0}{\text{ }}^{\mathbf {p} }\tau _{\geq 0}({\text{rat}}_{X}(M^{\bullet })[+k])}$

This kind of setup is also reflected in the derived push and pull functors ${\displaystyle f_{!},f^{*},f^{!},f_{*}}$ and with nearby and vanishing cycles ${\displaystyle \psi _{f},\phi _{f}}$, 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 ${\displaystyle p:X\to \{pt\}}$. One of the first mixed Hodge modules available is the weight 0 Tate object, denoted ${\displaystyle {\underline {\mathbb {Q} }}_{X}^{Hdg}}$ which is defined as the pullback of its corresponding object in ${\displaystyle \mathbb {Q} ^{Hdg}\in {\textbf {MHM}}(\{pt\})}$, so

${\displaystyle {\underline {\mathbb {Q} }}_{X}^{Hdg}=p^{*}\mathbb {Q} ^{Hdg}}$

The Hodge structure ${\displaystyle \mathbb {Q} ^{Hdg}}$ corresponds to the weight 0 Tate object ${\displaystyle \mathbb {Q} (0)}$ in the category of mixed Hodge structures. This object is useful because it can be used to compute the various cohomologies of ${\displaystyle X}$ through the six functor formalism and give them a mixed Hodge structure. These can be summarized with the table

${\displaystyle {\begin{matrix}H^{k}(X;\mathbb {Q} )&=H^{k}(\{pt\},p_{*}p^{*}\mathbb {Q} ^{Hdg})\\H_{c}^{k}(X;\mathbb {Q} )&=H^{k}(\{pt\},p_{!}p^{*}\mathbb {Q} ^{Hdg})\\H_{-k}(X;\mathbb {Q} )&=H^{k}(\{pt\},p_{!}p^{!}\mathbb {Q} ^{Hdg})\\H_{-k}^{BM}(X;\mathbb {Q} )&=H^{k}(\{pt\},p_{!}p^{*}\mathbb {Q} ^{Hdg})\end{matrix}}}$

Moreover, given a closed embedding ${\displaystyle i:Z\to X}$ there is the local cohomology group

${\displaystyle H_{Z}^{k}(X;\mathbb {Q} )=H^{k}(\{pt\},p_{*}i_{*}i^{!}{\underline {\mathbb {Q} }}_{X}^{Hdg})}$

For a morphism of varieties ${\displaystyle f:X\to Y}$ the pushforward maps ${\displaystyle f_{*}{\underline {\mathbb {Q} }}_{X}^{Hdg}}$ and ${\displaystyle f_{!}{\underline {\mathbb {Q} }}_{X}^{Hdg}}$ give degenerating variations of mixed Hodge structures on ${\displaystyle Y}$. 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 ${\displaystyle f:X\to Y}$ of varieties. To define the intersection complex, let ${\displaystyle j:U\hookrightarrow X}$ be the open smooth part of a variety ${\displaystyle X}$. Then the intersection complex of ${\displaystyle X}$ can be defined as

${\displaystyle IC_{X}^{\bullet }\mathbb {Q} ^{Hdg}:=j_{!*}{\underline {\mathbb {Q} }}_{U}^{Hdg}[d_{X}]}$

where

${\displaystyle j_{!*}({\underline {\mathbb {Q} }}_{U}^{Hdg})=\operatorname {Image} [j_{!}({\underline {\mathbb {Q} }}_{U}^{Hdg})\to j_{*}({\underline {\mathbb {Q} }}_{U}^{Hdg})]}$

as with perverse sheaves^{[2]pg 311}. In particular, this setup can be used to show the intersection cohomology groups

${\displaystyle IH^{k}(X)=H^{k}(p_{*}IC^{\bullet }{\underline {\mathbb {Q} }}_{X})}$

have a pure weight ${\displaystyle k}$ Hodge structure.

• Mixed motives (math)
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• A young person's guide to mixed Hodge modules