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. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).
Hodge structures.
Definition of Hodge structures.
A pure Hodge structure of integer weight "n" consists of an abelian group formula_1 and a decomposition of its complexification "H" into a direct sum of complex subspaces formula_2, where formula_3, with the property that the complex conjugate of formula_2 is formula_5:
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 formula_8 subject to the condition
The relation between these two descriptions is given as follows:
For example, if "X" is a compact Kähler manifold, formula_12 is the "n"-th cohomology group of "X" with integer coefficients, then formula_13 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 formula_14 with the decreasing filtration by formula_15 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 formula_1 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 formula_17 and a non-degenerate integer bilinear form "Q" on formula_1 (polarization), 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 formula_21 on formula_2.
Yet another definition of a Hodge structure is based on the equivalence between the formula_23-grading on a complex vector space and the action of the circle group U(1). In this definition, an action of the multiplicative group of complex numbers formula_24 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 formula_2 is the subspace on which formula_26 acts as multiplication by formula_27
"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 formula_28 of real numbers, for which formula_29 is a field. Then a pure Hodge A-structure of weight "n" is defined as before, replacing formula_23 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 (possibly reducible) 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 "P""X"("t"), 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 (singular and non-complete) algebraic variety. The novel feature is that the "n"th 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 (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.
Example of curves.
To motivate the definition, consider the case of a reducible complex algebraic curve "X" consisting of two nonsingular components, formula_33 and formula_34, which transversally intersect at the points formula_35 and formula_36. Further, assume that the components are not compact, but can be compactified by adding the points formula_37. The first cohomology group of the curve "X" (with compact support) 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 formula_38 representing small loops around the punctures formula_39. Then there are elements formula_40 that are coming from the first homology of the compactification of each of the components. The one-cycle in formula_41 (formula_42) corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of formula_43. Finally, modulo the first two types, the group is generated by a combinatorial cycle formula_44 which goes from formula_35 to formula_36along a path in one component formula_33 and comes back along a path in the other component formula_34. This suggests that formula_49 admits an increasing filtration
whose successive quotients "Wn"/"W""n"−1 originate from the cohomology of smooth complete varieties, hence admit (pure) 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 formula_1 consists of a finite decreasing filtration "Fp" on the complex vector space "H" (the complexification of formula_1), called the Hodge filtration and a finite increasing filtration "Wi" on the rational vector space formula_53 (obtained by extending the scalars to rational numbers), called the weight filtration, subject to the requirement that the "n"-th associated graded quotient of formula_54 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 "n"th space of the weight filtration "Wn" is the direct sum of the cohomology groups (with rational coefficients) 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 formula_57 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. 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 (much more involved) analysis for rational pure polarizable Hodge structures was done by .
Mixed Hodge structure in cohomology (Deligne's theorem).
Deligne has proved that the "n"th cohomology group of an arbitrary algebraic variety has a canonical mixed Hodge structure. This structure is functorial, and compatible with the products of varieties ("Künneth isomorphism") 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 (due to Hironaka) 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 (as opposed to cohomology) 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.
Applications.
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 formula_89 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" ⊗ "O""X", subject to the following two conditions:
Here the natural (flat) connection on "S" ⊗ "OX" induced by the flat connection on "S" and the flat connection "d" on "O""X", and "OX" is the sheaf of holomorphic functions on "X", and formula_92 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 formula_93. For example,
has fibers
which are smooth plane curves of genus 10 for formula_96 and degenerate to a singular curve at formula_97 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 (derived categories of) mixed Hodge modules similar to the ones for sheaves.