All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).
A pure Hodge structure of integer weight n consists of an abelian group
-th cohomology group with complex coefficients and Hodge theory provides the decomposition of
into a direct sum as above, so that these data define a pure Hodge structure of weight
[1] For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight
The definition of a Hodge structure is modified by fixing a Noetherian subring A of the field
There are natural functors of base change and restriction relating Hodge A-structures and B-structures for A a subring of B.
It was noticed by Jean-Pierre Serre in the 1960s based on the Weil conjectures that even singular (possibly reducible) and non-complete algebraic varieties should admit 'virtual Betti numbers'.
The novel feature is that the nth cohomology of a general variety looks as if it contained pieces of different weights.
He introduced the notion of a mixed Hodge structure, developed techniques for working with them, gave their construction (based on Heisuke Hironaka's resolution of singularities) and related them to the weights on l-adic cohomology, proving the last part of the Weil conjectures.
To motivate the definition, consider the case of a reducible complex algebraic curve X consisting of two nonsingular components,
) corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of
Further examples can be found in "A Naive Guide to Mixed Hodge Theory".
consists of a finite decreasing filtration Fp on the complex vector space H (the complexification of
(obtained by extending the scalars to rational numbers), called the weight filtration, subject to the requirement that the n-th associated graded quotient of
In general, the total cohomology space still has these two filtrations, but they no longer come from a direct sum decomposition.
An important insight of Deligne is that in the mixed case there is a more complicated noncommutative proalgebraic group that can be used to the same effect using Tannakian formalism.
Moreover, the category of (mixed) Hodge structures admits a good notion of tensor product, corresponding to the product of varieties, as well as related concepts of inner Hom and dual object, making it into a Tannakian category.
The corresponding (much more involved) analysis for rational pure polarizable Hodge structures was done by Patrikis (2016).
Deligne has proved that the nth cohomology group of an arbitrary algebraic variety has a canonical mixed Hodge structure.
For a complete nonsingular variety X this structure is pure of weight n, and the Hodge filtration can be defined through the hypercohomology of the truncated de Rham complex.
The proof roughly consists of two parts, taking care of noncompactness and singularities.
In the singular case, varieties are replaced by simplicial schemes, leading to more complicated homological algebra, and a technical notion of a Hodge structure on complexes (as opposed to cohomology) is used.
[5] The machinery based on the notions of Hodge structure and mixed Hodge structure forms a part of still largely conjectural theory of motives envisaged by Alexander Grothendieck.
Arithmetic information for nonsingular algebraic variety X, encoded by eigenvalue of Frobenius elements acting on its l-adic cohomology, has something in common with the Hodge structure arising from X considered as a complex algebraic variety.
Sergei Gelfand and Yuri Manin remarked around 1988 in their Methods of homological algebra, that unlike Galois symmetries acting on other cohomology groups, the origin of "Hodge symmetries" is very mysterious, although formally, they are expressed through the action of the fairly uncomplicated group
A variation of mixed Hodge structure can be defined in a similar way, by adding a grading or filtration W to S. Typical examples can be found from algebraic morphisms
Then, the cohomology sheaves give variations of mixed hodge structures.
They can be thought of informally as something like sheaves of Hodge structures on a manifold; the precise definition Saito (1989) is rather technical and complicated.
For each smooth complex variety, there is an abelian category of mixed Hodge modules associated with it.