Mixed Hodge structure

Note that similar to Hodge structures, mixed Hodge structures use a filtration instead of a direct sum decomposition since the cohomology groups with anti-holomorphic terms, ${\displaystyle H^{p,q}}$ where ${\displaystyle q>0}$, don't vary holomorphically. But, the filtrations can vary holomorphically, giving a better defined structure.

Morphisms of mixed Hodge structures are defined by maps of abelian groups

${\displaystyle f:(H_{\mathbb {Z} },W_{\bullet },F^{\bullet })\to (H_{\mathbb {Z} }',W_{\bullet }',F'^{\bullet })}$

such that

${\displaystyle f(W_{l})\subset W'_{l}}$

and the induced map of ${\displaystyle \mathbb {C} }$-vector spaces has the property

${\displaystyle f_{\mathbb {C} }(F^{p})\subset F'^{p}}$

The Hodge numbers of a MHS are defined as the dimensions

${\displaystyle h^{p,q}(H_{\mathbb {Z} })=\dim _{\mathbb {C} }{\text{Gr}}_{F^{\bullet }}^{p}{\text{Gr}}_{p+q}^{W_{\bullet }}H_{\mathbb {C} }}$

since ${\displaystyle {\text{Gr}}_{p+q}^{W_{\bullet }}H_{\mathbb {C} }}$ is a weight ${\displaystyle (p+q)}$ Hodge structure, and

${\displaystyle {\text{Gr}}_{p}^{F^{\bullet }}={\frac {F^{p}}{F^{p+1}}}}$

is the ${\displaystyle (p,q)}$-component of a weight ${\displaystyle (p+q)}$ Hodge structure.

There is an Abelian category[2] of mixed Hodge structures which has vanishing ${\displaystyle {\text{Ext}}}$-groups whenever the cohomological degree is greater than ${\displaystyle 1}$: that is, given mixed hodge structures ${\displaystyle (H_{\mathbb {Z} },W_{\bullet },F^{\bullet }),(H_{\mathbb {Z} }',W_{\bullet }',F'^{\bullet })}$ the groups

${\displaystyle \operatorname {Ext} _{MHS}^{p}((H_{\mathbb {Z} },W_{\bullet },F^{\bullet }),(H_{\mathbb {Z} }',W_{\bullet }',F'^{\bullet }))=0}$

for ${\displaystyle p\geq 2}$[2]^{pg 83}.

Given a smooth variety ${\displaystyle U\subset X}$ where ${\displaystyle D=X-U}$ is a normal crossing divisor (meaning all intersections of components are complete intersections), there are filtrations on the log cohomology complex ${\displaystyle \Omega _{X}^{\bullet }(\log D)}$ given by

${\displaystyle {\begin{aligned}W_{m}\Omega _{X}^{i}(\log D)&={\begin{cases}\Omega _{X}^{i}(\log D)&{\text{ if }}i\leq m\\\Omega _{X}^{i-m}\wedge \Omega _{X}^{m}(\log D)&{\text{ if }}0\leq m\leq i\\0&{\text{ if }}m<0\end{cases}}\\[6pt]F^{p}\Omega _{X}^{i}(\log D)&={\begin{cases}\Omega _{X}^{i}(\log D)&{\text{ if }}p\leq i\\0&{\text{ otherwise}}\end{cases}}\end{aligned}}}$

It turns out these filtrations define a natural mixed Hodge structure on the cohomology group ${\displaystyle H^{n}(U,\mathbb {C} )}$ from the mixed Hodge complex defined on the logarithmic complex ${\displaystyle \Omega _{X}^{\bullet }(\log D)}$.

The above construction of the logarithmic complex extends to every smooth variety; and the mixed Hodge structure is isomorphic under any such compactificaiton. Note a smooth compactification of a smooth variety ${\displaystyle U}$ is defined as a smooth variety ${\displaystyle X}$ and an embedding ${\displaystyle U\hookrightarrow X}$ such that ${\displaystyle D=X-U}$ is a normal crossing divisor. That is, given compactifications ${\displaystyle U\subset X,X'}$ with boundary divisors ${\displaystyle D=X-U,{\text{ }}D'=X'-U}$there is an isomorphism of mixed Hodge structure

${\displaystyle (\mathbb {H} ^{k}(X,\Omega _{X}^{\bullet }(\log D)),W_{\bullet },F^{\bullet })}$ ${\displaystyle (\mathbb {H} ^{k}(X',\Omega _{X'}^{\bullet }(\log D')),W_{\bullet },F^{\bullet })}$

showing the mixed Hodge structure is invariant under smooth compactification.[2]

For example, on a genus ${\displaystyle 0}$ plane curve ${\displaystyle C}$ logarithmic cohomology of ${\displaystyle C}$ with the normal crossing divisor ${\displaystyle \{p_{1},\ldots ,p_{k}\}}$ with ${\displaystyle k\geq 1}$ can be easily computed[3] since the terms of the complex ${\displaystyle \Omega _{C}^{\bullet }(\log D)}$ equal to

${\displaystyle {\mathcal {O}}_{C}\xrightarrow {d} \Omega _{C}^{1}(\log D)}$

are both acyclic. Then, the Hypercohomology is just

${\displaystyle \Gamma ({\mathcal {O}}_{\mathbb {P} ^{1}})\xrightarrow {d} \Gamma (\Omega _{\mathbb {P} ^{1}}(\log D))}$

the first vector space are just the constant sections, hence the differential is the zero map. The second is the vector space is isomorphic to the vector space spanned by

${\displaystyle \mathbb {C} \cdot {\frac {dx}{x-p_{1}}}\oplus \cdots \oplus \mathbb {C} {\frac {dx}{x-p_{k-1}}}}$

Then ${\displaystyle \mathbb {H} ^{1}(\Omega _{C}^{1}(\log D))}$ has a weight ${\displaystyle 2}$ mixed Hodge structure and ${\displaystyle \mathbb {H} ^{0}(\Omega _{C}^{1}(\log D))}$ has a weight ${\displaystyle 0}$ mixed Hodge structure.

Given a smooth projective variety ${\displaystyle X}$ of dimension ${\displaystyle n}$ and a closed subvariety ${\displaystyle Y\subset X}$ there is a long exact sequence in cohomology[4]^pg7-8

${\displaystyle \cdots \to H_{c}^{m}(U;\mathbb {Z} )\to H^{m}(X;\mathbb {Z} )\to H^{m}(Y;\mathbb {Z} )\to H_{c}^{m+1}(U;\mathbb {Z} )\to \cdots }$

coming from the distinguished triangle

${\displaystyle \mathbf {R} j_{!}\mathbb {Z} _{U}\to \mathbb {Z} _{X}\to i_{*}\mathbb {Z} _{Y}\xrightarrow {[+1]} }$

of constructible sheaves. There is another long exact sequence

${\displaystyle \cdots \to H_{2n-m}^{BM}(Y;\mathbb {Z} )(-n)\to H^{m}(X;\mathbb {Z} )\to H^{m}(U;\mathbb {Z} )\to H_{2n-m-1}^{BM}(U;\mathbb {Z} )(-n)\to \cdots }$

from the distinguished triangle

${\displaystyle i_{*}i^{!}\mathbb {Z} _{X}\to \mathbb {Z} _{X}\to \mathbf {R} j_{*}\mathbb {Z} _{U}\xrightarrow {[+1]} }$

whenever ${\displaystyle X}$ is smooth. Note the homology groups ${\displaystyle H_{k}^{BM}(X)}$ are called Borel–Moore homology, which are dual to cohomology for general spaces and the ${\displaystyle (n)}$ means tensoring with the Tate structure ${\displaystyle \mathbb {Z} (1)^{\otimes n}}$ add weight ${\displaystyle -2n}$ to the weight filtration. The smoothness hypothesis is required because Verdier duality implies ${\displaystyle i^{!}D_{X}=D_{Y}}$, and ${\displaystyle D_{X}\cong \mathbb {Z} _{X}[2n]}$ whenever ${\displaystyle X}$ is smooth. Also, the dualizing complex for ${\displaystyle X}$ has weight ${\displaystyle n}$, hence ${\displaystyle D_{X}\cong \mathbb {Z} _{X}[2n](n)}$. Also, the maps from Borel-Moore homology must be twisted by up to weight ${\displaystyle (n)}$ is order for it to have a map to ${\displaystyle H^{m}(X)}$. Also, there is the perfect duality paring

${\displaystyle H_{2n-m}^{BM}(Y)\times H^{m}(Y)\to \mathbb {Z} }$

giving an isomophism of the two groups.

A one dimensional algebraic torus ${\displaystyle \mathbb {T} }$ is isomorphic to the variety ${\displaystyle \mathbb {P} ^{1}-\{0,\infty \}}$, hence its cohomology groups are isomorphic to

${\displaystyle {\begin{aligned}H^{0}(\mathbb {T} )\oplus H^{1}(\mathbb {T} )&\cong \mathbb {Z} \oplus \mathbb {Z} \end{aligned}}}$

The long exact exact sequence then reads

${\displaystyle {\begin{matrix}&H_{0}^{BM}(Y)(1)\to H^{2}(\mathbb {P} ^{1})\to H^{2}(\mathbb {G} _{m})\to 0\\&H_{1}^{BM}(Y)(1)\to H^{1}(\mathbb {P} ^{1})\to H^{1}(\mathbb {G} _{m})\to {\text{ }}\\&H_{2}^{BM}(Y)(1)\to H^{0}(\mathbb {P} ^{1})\to H^{0}(\mathbb {G} _{m})\to {\text{ }}\\\end{matrix}}}$

Since ${\displaystyle H^{1}(\mathbb {P} ^{1})=0}$ and ${\displaystyle H^{2}(\mathbb {G} _{m})=0}$ this gives the exact sequence

${\displaystyle 0\to H^{1}(\mathbb {G} _{m})\to H_{0}^{BM}(Y)(1)\to H^{2}(\mathbb {P} ^{1})\to 0}$

since there is a twisting of weights for well-defined maps of mixed Hodge structures, there is the isomorphism

${\displaystyle H^{1}(\mathbb {G} _{m})\cong \mathbb {Z} (-1)}$

Given a quartic K3 surface ${\displaystyle X}$, and a genus 3 curve ${\displaystyle i:C\hookrightarrow X}$ defined by the vanishing locus of a generic section of ${\displaystyle {\mathcal {O}}_{X}(1)}$, hence it is isomorphic to a degree ${\displaystyle 4}$ plane curve, which has genus 3. Then, the Gysin sequence gives the long exact sequence

${\displaystyle \to H^{k-2}(C)\xrightarrow {\gamma _{k}} H^{k}(X)\xrightarrow {i^{*}} H^{k}(U)\xrightarrow {R} H^{k-1}(C)\to }$

But, it is a result that the maps ${\displaystyle \gamma _{k}}$ take a Hodge class of type ${\displaystyle (p,q)}$ to a Hodge class of type ${\displaystyle (p+1,q+1)}$.[5] The Hodge structures for both the K3 surface and the curve are well-known, and can be computed using the Jacobian ideal. In the case of the curve there are two zero maps

${\displaystyle \gamma _{3}:H^{1,0}(C)\to H^{2,1}(X)=0}$ ${\displaystyle \gamma _{3}:H^{0,1}(C)\to H^{1,2}(X)=0}$

hence ${\displaystyle H^{2}(U)}$ contains the weight one pieces ${\displaystyle H^{1,0}(C)\oplus H^{0,1}(C)}$. Because ${\displaystyle H^{2}(X)=H_{\text{prim}}^{2}(X)\oplus \mathbb {C} \cdot \mathbb {L} }$ has dimension ${\displaystyle 22}$, but the Leftschetz class ${\displaystyle \mathbb {L} }$ is killed off by the map

${\displaystyle \gamma _{2}:H^{0}(C)\to H^{2}(X)}$

sending the ${\displaystyle (0,0)}$ class in ${\displaystyle H^{0}(C)}$ to the ${\displaystyle (1,1)}$ class in ${\displaystyle H^{2}(X)}$. Then the primitive cohomology group ${\displaystyle H_{\text{prim}}^{2}(X)}$ is the weight 2 piece of ${\displaystyle H^{2}(U)}$. Therefore,

${\displaystyle {\begin{aligned}{\text{Gr}}_{2}^{W_{\bullet }}H^{2}(U)&=H_{\text{prim}}^{2}(X)\\{\text{Gr}}_{1}^{W_{\bullet }}H^{2}(U)&=H^{1}(C)\\{\text{Gr}}_{k}^{W_{\bullet }}H^{2}(U)&=0&k\neq 1,2\end{aligned}}}$

The induced filtrations on these graded pieces are the Hodge filtrations coming from each cohomology group.

• A Naive Guide to Mixed Hodge Theory
• Introduction to Limit Mixed Hodge Structures
• Deligne’s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities

• Local B-Model and Mixed Hodge Structure