Hodge structure


In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. A mixed Hodge structure is a generalization, defined by Pierre Deligne, that applies to all complex varieties. A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths. All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito.

Hodge structures

Definition of Hodge structures

A pure Hodge structure of integer weight n consists of an abelian group and a decomposition of its complexification H into a direct sum of complex subspaces Hp,q, where p + q = n, with the property that the complex conjugate of Hp,q is Hq,p:
An equivalent definition is obtained by replacing the direct sum decomposition of H by the Hodge filtration, a finite decreasing filtration of H by complex subspaces subject to the condition
The relation between these two descriptions is given as follows:
For example, if X is a compact Kähler manifold, is the n-th cohomology group of X with integer coefficients, then is its n-th cohomology group with complex coefficients and Hodge theory provides the decomposition of H into a direct sum as above, so that these data define a pure Hodge structure of weight n. On the other hand, the Hodge–de Rham spectral sequence supplies Hn with the decreasing filtration by FpH as in the second definition.
For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight n on is too big. Using the Riemann bilinear relations, in this case called Hodge Riemann bilinear relations, it can be substantially simplified. A polarized Hodge structure of weight n consists of a Hodge structure and a non-degenerate integer bilinear form Q on , which is extended to H by linearity, and satisfying the conditions:
In terms of the Hodge filtration, these conditions imply that
where C is the Weil operator on H, given by C = ip−q on Hp,q.
Yet another definition of a Hodge structure is based on the equivalence between the -grading on a complex vector space and the action of the circle group U. In this definition, an action of the multiplicative group of complex numbers viewed as a two-dimensional real algebraic torus, is given on H. This action must have the property that a real number a acts by an. The subspace Hp,q is the subspace on which acts as multiplication by

''A''-Hodge structure

In the theory of motives, it becomes important to allow more general coefficients for the cohomology. The definition of a Hodge structure is modified by fixing a Noetherian subring A of the field of real numbers, for which is a field. Then a pure Hodge A-structure of weight n is defined as before, replacing with A. There are natural functors of base change and restriction relating Hodge A-structures and B-structures for A a subring of B.

Mixed Hodge structures

It was noticed by Jean-Pierre Serre in the 1960s based on the Weil conjectures that even singular and non-complete algebraic varieties should admit 'virtual Betti numbers'. More precisely, one should be able to assign to any algebraic variety X a polynomial PX, called its virtual Poincaré polynomial, with the properties
The existence of such polynomials would follow from the existence of an analogue of Hodge structure in the cohomologies of a general algebraic variety. The novel feature is that the nth cohomology of a general variety looks as if it contained pieces of different weights. This led Alexander Grothendieck to his conjectural theory of motives and motivated a search for an extension of Hodge theory, which culminated in the work of Pierre Deligne. He introduced the notion of a mixed Hodge structure, developed techniques for working with them, gave their construction and related them to the weights on l-adic cohomology, proving the last part of the Weil conjectures.

Example of curves

To motivate the definition, consider the case of a reducible complex algebraic curve X consisting of two nonsingular components X1 and X2, which transversally intersect at the points Q1 and Q2. Further, assume that the components are not compact, but can be compactified by adding the points P1,..., Pn. The first cohomology group of the curve X is dual to the first homology group, which is easier to visualize. There are three types of one-cycles in this group. First, there are elements αi representing small loops around the punctures Pi. Then there are elements βj that are coming from the first homology of the compactification of each of the components. The one-cycle in,
k = 1, 2, corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of αi. Finally, modulo the first two types, the group is generated by a combinatorial cycle γ which goes from Q1 to Q2 along a path in one component X1 and comes back along a path in the other component X2. This suggests that H1 admits an increasing filtration
whose successive quotients Wn/Wn−1 originate from the cohomology of smooth complete varieties, hence admit Hodge structures, albeit of different weights. Further examples can be found in "A Naive Guide to Mixed Hodge Theory".

Definition of mixed Hodge structure

A mixed Hodge structure on an abelian group consists of a finite decreasing filtration Fp on the complex vector space H, called the Hodge filtration and a finite increasing filtration Wi on the rational vector space , called the weight filtration, subject to the requirement that the n-th associated graded quotient of with respect to the weight filtration, together with the filtration induced by F on its complexification, is a pure Hodge structure of weight n, for all integer n. Here the induced filtration on
is defined by
One can define a notion of a morphism of mixed Hodge structures, which has to be compatible with the filtrations F and W and prove the following:
The total cohomology of a compact Kähler manifold has a mixed Hodge structure, where the nth space of the weight filtration Wn is the direct sum of the cohomology groups of degree less than or equal to n. Therefore, one can think of classical Hodge theory in the compact, complex case as providing a double grading on the complex cohomology group, which defines an increasing fitration Fp and a decreasing filtration Wn that are compatible in certain way. In general, the total cohomology space still has these two filtrations, but they no longer come from a direct sum decomposition. In relation with the third definition of the pure Hodge structure, one can say that a mixed Hodge structure cannot be described using the action of the group 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 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. By Tannaka–Krein philosophy, this category is equivalent to the category of finite-dimensional representations of a certain group, which Deligne, Milne and et el. has explicitly described, see and. The description of this group was recast in more geometrical terms by. The corresponding analysis for rational pure polarizable Hodge structures was done by.

Mixed Hodge structure in cohomology (Deligne's theorem)

Deligne has proved that the nth cohomology group of an arbitrary algebraic variety has a canonical mixed Hodge structure. This structure is functorial, and compatible with the products of varieties and the product in cohomology. 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. Both parts use the resolution of singularities in an essential way. 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 is used.
Using the theory of motives, it is possible to refine the weight filtration on the cohomology with rational coefficients to one with integral coefficients.

Examples

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 on the de Rham cohomology. Since then, the mystery has deepened with the discovery and mathematical formulation of mirror symmetry.

Variation of Hodge structure

A variation of Hodge structure is a family of Hodge structures
parameterized by a complex manifold X. More precisely a variation of Hodge structure of weight n on a complex manifold X consists of a locally constant sheaf S of finitely generated abelian groups on X, together with a decreasing Hodge filtration F on S ⊗ OX, subject to the following two conditions:
Here the natural connection on S ⊗ OX induced by the flat connection on S and the flat connection d on OX, and OX is the sheaf of holomorphic functions on X, and is the sheaf of 1-forms on X. This natural flat connection is a Gauss–Manin connection ∇ and can be described by the Picard–Fuchs equation.
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. For example,
has fibers
which are smooth plane curves of genus 10 for and degenerate to a singular curve at Then, the cohomology sheaves
give variations of mixed hodge structures.

Hodge modules

Hodge modules are a generalization of variation of Hodge structures on a complex manifold. They can be thought of informally as something like sheaves of Hodge structures on a manifold; the precise definition is rather technical and complicated. There are generalizations to mixed Hodge modules, and to manifolds with singularities.
For each smooth complex variety, there is an abelian category of mixed Hodge modules associated with it. These behave formally like the categories of sheaves over the manifolds; for example, morphisms f between manifolds induce functors f∗, f*, f!, f! between mixed Hodge modules similar to the ones for sheaves.

Introductory references

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