Mixed Hodge structure

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 ${\displaystyle H^{k}(X)}$ may have subspaces of different weights, i.e. as a direct sum of Hodge structures

${\displaystyle H^{k}(X)=\bigoplus _{i}(H_{i},F_{i}^{\bullet })}$

where each of the Hodge structures have weight ${\displaystyle k_{i}}$. One of the early hints that such structures should exist comes from the long exact sequence ${\displaystyle \dots \to H^{i-1}(Y)\to H_{c}^{i}(U)\to H^{i}(X)\to \dots }$associated to a pair of smooth projective varieties ${\displaystyle Y\subset X}$. This sequence suggests that the cohomology groups ${\displaystyle H_{c}^{i}(U)}$ (for ${\displaystyle U=X-Y}$) should have differing weights coming from both ${\displaystyle H^{i-1}(Y)}$ and ${\displaystyle H^{i}(X)}$.