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
may have subspaces of different weights, i.e. as a direct sum of Hodge structures where each of the Hodge structures have weight
One of the early hints that such structures should exist comes from the long exact sequence
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.
A mixed Hodge structure[1] (MHS) is a triple
are pure Hodge structures of weight
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,
But, the filtrations can vary holomorphically, giving a better defined structure.
Morphisms of mixed Hodge structures are defined by maps of abelian groups
The Hodge numbers of a MHS are defined as the dimensions
There is an Abelian category[2] of mixed Hodge structures which has vanishing
Many mixed Hodge structures can be constructed from a bifiltered complex.
Given a complex of sheaves of abelian groups
There is an induced mixed Hodge structure on the hyperhomology groups
is a normal crossing divisor (meaning all intersections of components are complete intersections), there are filtrations on the logarithmic de Rham complex
It turns out these filtrations define a natural mixed Hodge structure on the cohomology group
The above construction of the logarithmic complex extends to every smooth variety; and the mixed Hodge structure is isomorphic under any such compactificaiton.
showing the mixed Hodge structure is invariant under smooth compactification.
can be easily computed[3] since the terms of the complex
the first vector space are just the constant sections, hence the differential is the zero map.
are called Borel–Moore homology, which are dual to cohomology for general spaces and the
The smoothness hypothesis is required because Verdier duality implies
Also, the maps from Borel-Moore homology must be twisted by up to weight
since there is a twisting of weights for well-defined maps of mixed Hodge structures, there is the isomorphism
defined by the vanishing locus of a generic section of
[5] The Hodge structures for both the K3 surface and the curve are well-known, and can be computed using the Jacobian ideal.