Mixed Hodge Structures

Hodge theory tells us that the cohomology of a compact Kähler manifold is a pure Hodge structure. What if we consider an algebraic variety (over $\mathbb {C}$$\mathbb {C}$) in general? The question whether its cohomology should have a pure Hodge structure has negative answer. Consider a nodal elliptic curve $C$$C$. It turns out that $H^{1}(C)\simeq \mathbb {Z} \langle dy\rangle$$H^{1}(C)\simeq \mathbb {Z} \langle dy\rangle$, hence it does not admit a complex structure and, in particular, is not a pure Hodge structure of weight 1. If we resolve the singularity, we get ${\bar {C}}$${\bar {C}}$ with exceptional divisor $E$$E$ such that $C={\bar {C}}/E$$C={\bar {C}}/E$ and we have a long exact sequence of relative homology groups

$$0\longleftarrow {\tilde {H}}_{0}(E)\longleftarrow H_{1}({\bar {C}},E)\longleftarrow H_{1}({\bar {C}})\longleftarrow H_{1}(E)=0\longleftarrow \cdots ,$$

$$0\longleftarrow {\tilde {H}}_{0}(E)\longleftarrow H_{1}({\bar {C}},E)\longleftarrow H_{1}({\bar {C}})\longleftarrow H_{1}(E)=0\longleftarrow \cdots ,$$

and the relative exact sequence in cohomology

$$0\longrightarrow {\tilde {H}}^{0}(E)\longrightarrow H^{1}({\bar {C}},E)\longrightarrow H^{1}({\bar {C}})\longrightarrow 0.$$

$$0\longrightarrow {\tilde {H}}^{0}(E)\longrightarrow H^{1}({\bar {C}},E)\longrightarrow H^{1}({\bar {C}})\longrightarrow 0.$$

Since $H^{1}(C)\simeq H^{1}({\bar {C}},E)$$H^{1}(C)\simeq H^{1}({\bar {C}},E)$, we can see $H^{1}(C)$$H^{1}(C)$ as the extension of $H^{0}(E)$$H^{0}(E)$ and $H^{1}({\bar {C}})$$H^{1}({\bar {C}})$, which are, respectively, pure Hodge structures of weight 0 and 1. More precisely, we have an increasing filtration

$$\operatorname {im} H^{0}(E)=W^{0}\subset W^{1}=H^{1}(C),$$

$$\operatorname {im} H^{0}(E)=W^{0}\subset W^{1}=H^{1}(C),$$

such that the graded piece

$$\operatorname {Gr_{1}^{W}} :=qH^{1}(C)/{\tilde {H}}^{0}(E)\simeq H^{1}({\bar {C}})$$

$$\operatorname {Gr_{1}^{W}} :=qH^{1}(C)/{\tilde {H}}^{0}(E)\simeq H^{1}({\bar {C}})$$

is a pure Hodge structure of weight 1.

Definition 11.4

A mixed Hodge structure $(H_{\mathbb {Z} },W^{\bullet },F^{\bullet })$$(H_{\mathbb {Z} },W^{\bullet },F^{\bullet })$ consists of a free abelian group $H_{\mathbb {Z} }$$H_{\mathbb {Z} }$ together with an increasing filtration of $H_{\mathbb {Q} }$$H_{\mathbb {Q} }$

$$W^{0}\subseteq W^{1}\subseteq W^{2}\subseteq \cdots ,$$

$$W^{0}\subseteq W^{1}\subseteq W^{2}\subseteq \cdots ,$$

and a dicreasing filtration of $H_{\mathbb {C} }$$H_{\mathbb {C} }$

$$H_{\mathbb {C} }=F^{0}\supset F^{1}\supset F^{2}\supset \cdots ,$$

$$H_{\mathbb {C} }=F^{0}\supset F^{1}\supset F^{2}\supset \cdots ,$$

such that $F^{\bullet }$$F^{\bullet }$ defines a pure Hodge structure of weight $k$$k$ on the graded piece

$$\operatorname {Gr_{k}^{W}} H_{\mathbb {C} }=W^{k}/W^{k-1}.$$

$$\operatorname {Gr_{k}^{W}} H_{\mathbb {C} }=W^{k}/W^{k-1}.$$

 

A morphism of mixed Hodge structures is a $\mathbb {Z}$$\mathbb{Z}$-linear map which is compatible with the two filtrations of filtered vector spaces.

Proposition 11.1

Any morphism $f:(V_{\mathbb {Z} },W,F)\to (V'_{\mathbb {Z} },W',F')$$f:(V_{\mathbb {Z} },W,F)\to (V'_{\mathbb {Z} },W',F')$ of mixed Hodge structures is strict, i.e. every element of $F'^{p}$$F'^{p}$ which is in the image of $f$$f$ comes from $F^{p}$$F^{p}$ and similarly the same holds for the weight filtration.

 

In view of the (singular) case above, we would like to have a (canonical and functorial) result independent of resolution and compactification, which allows us to associate a mixed Hodge structure to any algebraic variety over $\mathbb {C}$$\mathbb {C}$. Indeed, we have the following.

Theorem 11.1

Quasi-projective varieties have canonical mixed Hodge structure on their cohomology.

 

Moreover, we get that the filtrations are independent of the choice of resolution, and so, by Hironaka's theorem, the result is extended to any algebraic variety over $\mathbb {C}$$\mathbb {C}$.