In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. In mixed Hodge theory, where the decomposition of a cohomology group H k ( X ) {\displaystyle H^{k}(X)} may have subspaces of different weights, i.e. as a direct sum of Hodge structures
H k ( X ) = ⨁ i ( H i , F i ∙ ) {\displaystyle H^{k}(X)=\bigoplus _{i}(H_{i},F_{i}^{\bullet })}
where each of the Hodge structures have weight k i {\displaystyle k_{i}} . One of the early hints that such structures should exist comes from the long exact sequence ⋯ → H i − 1 ( Y ) → H c i ( U ) → H i ( X ) → … {\displaystyle \dots \to H^{i-1}(Y)\to H_{c}^{i}(U)\to H^{i}(X)\to \dots } associated to a pair of smooth projective varieties Y ⊂ X {\displaystyle Y\subset X} . This sequence suggests that the cohomology groups H c i ( U ) {\displaystyle H_{c}^{i}(U)} (for U = X − Y {\displaystyle U=X-Y} ) should have differing weights coming from both H i − 1 ( Y ) {\displaystyle H^{i-1}(Y)} and H i ( X ) {\displaystyle H^{i}(X)} .
Motivation Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups of smooth projective algebraic varieties. These structures gave geometers new tools for studying algebraic curves, such as the Torelli theorem, Abelian varieties, and the cohomology of smooth projective varieties. One of the chief results for computing Hodge structures is an explicit decomposition of the cohomology groups of smooth hypersurfaces using the relation between the Jacobian ideal and the Hodge decomposition of a smooth projective hypersurface through Griffith's residue theorem. Porting this language to smooth non-projective varieties and singular varieties requires the concept of mixed Hodge structures.
Definition A mixed Hodge structure (MHS) is a triple ( H Z , W ∙ , F ∙ ) {\displaystyle (H_{\mathbb {Z} },W_{\bullet },F^{\bullet })} such that
H Z {\displaystyle H_{\mathbb {Z} }} is a Z {\displaystyle \mathbb {Z} } -module of finite type
W ∙ {\displaystyle W_{\bullet }} is an increasing Z {\displaystyle \mathbb {Z} } -filtration on H Q = H Z ⊗ Q {\displaystyle H_{\mathbb {Q} }=H_{\mathbb {Z} }\otimes \mathbb {Q} } , ⋯ ⊂ W 0 ⊂ W 1 ⊂ W 2 ⊂ ⋯ {\displaystyle \cdots \subset W_{0}\subset W_{1}\subset W_{2}\subset \cdots }
F ∙ {\displaystyle F^{\bullet }} is a decreasing N {\displaystyle \mathbb {N} } -filtration on H C {\displaystyle H_{\mathbb {C} }} , H C = F 0 ⊃ F 1 ⊃ F 2 ⊃ ⋯ {\displaystyle H_{\mathbb {C} }=F^{0}\supset F^{1}\supset F^{2}\supset \cdots }
… excerpt ends here. Continue reading the full article.
