In mathematics, a Hodge structure, named after
W. V. D. Hodge, is an algebraic structure at the level of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, similar to the one that
Hodge theory gives to the
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
s of a smooth and compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
. Hodge structures have been generalized for all complex varieties (even if they are
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
and
non-complete) in the form of
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.
...
s, defined by
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
(1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by
Phillip Griffiths
Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
(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
and a decomposition of its complexification ''H'' into a direct sum of complex subspaces
, where
, with the property that the complex conjugate of
is
:
:
:
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
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
,
is the ''n''-th
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
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
with the decreasing filtration by
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 mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:
* A lattice Λ in a complex vector space Cg.
* An alternating bilinear form α from Λ to the integers satisfying the following Riemann bi ...
, 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
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
''Q'' on
(
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
on
.
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(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
. 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 ''a
n''. The subspace
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 In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
subring A of the field
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, 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
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
in the 1960s based on the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
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
* If ''X'' is nonsingular and projective (or complete)
* If ''Y'' is closed algebraic subset of ''X'' and ''U'' = ''X'' \ ''Y''
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
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
. 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
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
, proving the last part of the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
.
Example of curves
To motivate the definition, consider the case of a reducible complex
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
''X'' consisting of two nonsingular components,
and
, which transversally intersect at the points
and
. Further, assume that the components are not compact, but can be compactified by adding the points
. 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
representing small loops around the punctures
. Then there are elements
that are coming from the first homology of the
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
of each of the components. The one-cycle in
(
) corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of
. Finally, modulo the first two types, the group is generated by a combinatorial cycle
which goes from
to
along a path in one component
and comes back along a path in the other component
. This suggests that
admits an increasing filtration
:
whose successive quotients ''W
n''/''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
consists of a finite decreasing filtration ''F
p'' on the complex vector space ''H'' (the complexification of
), called the Hodge filtration and a finite increasing filtration ''W
i'' on the rational vector space
(obtained by extending the scalars to rational numbers), 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:
:Theorem. ''Mixed Hodge structures form an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
. The kernels and cokernels in this category coincide with the usual kernels and cokernels in the category of vector spaces, with the induced filtrations.''
The total cohomology of a compact Kähler manifold has a mixed Hodge structure, where the ''n''th space of the weight filtration ''W
n'' 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 ''F
p'' and a decreasing filtration ''W
n'' 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
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...
.
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
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...
. 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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
, 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
In homological algebra, the hyperhomology or hypercohomology (\mathbb_*(-), \mathbb^*(-)) is a generalization of (co)homology functors which takes as input not objects in an abelian category \mathcal but instead chain complexes of objects, so objec ...
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.
Examples
*The Tate–Hodge structure
is the Hodge structure with underlying
module given by
(a subgroup of
), with
So it is pure of weight −2 by definition and it is the unique 1-dimensional pure Hodge structure of weight −2 up to isomorphisms. More generally, its ''n''th tensor power is denoted by
it is 1-dimensional and pure of weight −2''n''.
*The cohomology of a complete Kähler manifold is a Hodge structure, and the subspace consisting of the ''n''th cohomology group is pure of weight ''n''.
*The cohomology of a complex variety (possibly singular or incomplete) is a mixed Hodge structure. This was shown for smooth varieties by , and in general by .
*For a projective variety
with
normal crossing singularities there is a spectral sequence with a degenerate E
2-page which computes all of its mixed Hodge structures. The E
1-page has explicit terms with a differential coming from a simplicial set.
*Any smooth affine variety admits a smooth compactification (which can be found taking its projective closure and finding its resolution of singularities) with a normal crossing divisor. The corresponding logarithmic forms can be used to find an explicit weight filtration of the mixed Hodge structure.
*The Hodge structure for a smooth projective hypersurface
of degree
was worked out explicitly by Griffiths in his "Period Integrals of Algebraic Manifolds" paper. If