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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 group]]s of a smooth and compact [[Kähler manifold]]. A '''mixed Hodge structure''' is a generalization, defined by [[Pierre Deligne]] (1970), that applies to all complex varieties (even if they are [[Mathematical singularity|singular]] and [[complete variety|non-complete]]). A '''variation of Hodge structure''' is a family of Hodge structures parameterized by a manifold, first studied by [[P. A. Griffiths]] (1968). All these concepts were further generalized to '''mixed Hodge modules''' over complex varieties by M. Saito (1989).
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==Hodge structures==
===Definition of Hodge structures===
'''A pure Hodge structure of weight ''n'' ''' (''n''&nbsp;∈&nbsp;'''Z''') consists of an abelian group ''H''<sub>'''Z'''</sub> and a decomposition of its complexification ''H'' 
into a direct sum of complex subspaces ''H''<sup>''p'',''q''</sup>, where ''p''&nbsp;+&nbsp;''q''&nbsp;=&nbsp;''n'', with the property that the complex conjugate of ''H''<sup>''p'',''q''</sup> is ''H''<sup>''q'',''p''</sup>:
 
:<math>H := H_{\mathbf{Z}}\otimes_{\mathbf Z} {\mathbf C} = \bigoplus\nolimits_{p+q=n}H^{p,q},</math>
:<math>\overline{H^{p,q}}=H^{q,p}.</math>
 
An equivalent definition is obtained by replacing the direct sum decomposition of ''H'' by the '''Hodge filtration''', a finite decreasing [[filtration (mathematics)|filtration]] of ''H'' by complex subspaces ''F''<sup>''p''</sup>''H'' (''p''&nbsp;∈&nbsp;'''Z'''), subject to the condition
 
: <math>\forall p, q \ : \ p + q = n+1 \ \ F^p H\cap\overline{F^q H}=0 </math>
 
The relation between these two descriptions is given as follows:
 
:<math> H^{p,q}=F^p H\cap \overline{F^q H},</math>
:<math>F^p H= \bigoplus\nolimits_{i\geq p} H^{i,n-i}. </math>
 
For example, if ''X'' is a compact [[Kähler manifold]], ''H''<sub>'''Z'''</sub>&nbsp;=&nbsp;''H''<sup>''n''</sup>(''X'', '''Z''')</sub> is the ''n''th [[cohomology group]] of ''X'' with integer coefficients, then ''H''&nbsp;=&nbsp;''H''<sup>''n''</sup>(''X'',&nbsp;'''C''') 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 ''H<sup>n</sup>'' with the decreasing filtration by ''F<sup>p</sup>H'' as in the second definition. <ref>In terms of spectral sequences, see [[homological algebra]], Hodge fitrations can be described as the following:
:<math>E^{p,q}_1=H^{p+q}(gr^W_nH)\Rightarrow H^{p+q},</math>
using notations in [[#Definition of mixed Hodge structure]]. The important fact is that this is degenerate at the term ''E''<sup>1</sup>, which means the Hodge-de Rham spectral sequence, and then the Hodge decomposition, depends only on the complex structure not Kahler metric on ''M''.</ref>
 
For applications in algebraic geometry, namely, classification of complex projective varieties by their [[period mapping|periods]], the set of all Hodge structures of weight ''n'' on ''H''<sub>'''Z'''</sub> is too big. Using the [[Riemann bilinear relations]], in this case called ''Hodge Riemann bilinear relations'', it can be substantially much simplified. A '''polarized Hodge structure of weight ''n'' ''' consists of a Hodge structure (''H''<sub>'''Z'''</sub>,  ''H''<sup>''p'',''q''</sup>) and a nondegenerate integer [[bilinear form]] ''Q'' on ''H''<sub>'''Z'''</sub> ([[abelian variety#Polarisation and dual abelian variety|polarization]]), which is extended to ''H'' by linearity, and satisfying the conditions:
 
:<math>\begin{align}
Q(\varphi,\psi) &= (-1)^n Q(\psi, \varphi); \\
Q(\varphi,\psi) &=0 && \text{ for }\varphi\in H^{p,q}, \psi\in H^{p',q'}, p\ne q'; \\
i^{p-q}Q \left(\varphi,\bar{\varphi} \right) &>0 && \text{ for }\varphi\in H^{p,q},\ \varphi\ne 0.
\end{align}</math>
 
In terms of the Hodge filtration, these conditions imply that
 
:<math>\begin{align}
Q\left (F^p, F^{n-p+1} \right ) &=0, \\
Q \left (C\varphi,\bar{\varphi} \right ) &>0 && \text{ for }\varphi\ne 0,
\end{align}</math>
 
where ''C'' is the ''Weil operator'' on ''H'', given by ''C'' = ''i''<sup>''p''−''q''</sup> on ''H''<sup>''p'',''q''</sup>.
 
Yet another definition of a Hodge structure is based on the equivalence between the '''Z'''-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 '''C'''*, viewed as a two-dimensional real algebraic  torus, is given on ''H''.<ref>More precisely, let ''S'' be the two-dimensional commutative real [[algebraic group]] defined as the [[Weil restriction]] of the [[multiplicative group]] from '''C''' to '''R'''; in other words, if ''A'' is an algebra over '''R''', then the group ''S''(''A'') of ''A''-valued points of ''G'' is the multiplicative group of ''A'' ⊗ '''C'''. Then ''S''('''R''') is the group '''C'''* of non-zero complex numbers.</ref> This action must have the property that a real number ''a'' acts by ''a''<sup>''n''</sup>. The subspace ''H''<sup>''p'',''q''</sup> is the subspace on which ''z''&nbsp;∈&nbsp;'''C'''* acts as multiplication by <math>z^p\overline{z}^q.</math>
 
<!--- Weight filtration is part of mixed Hodge structure The '''weight filtration'''  ''W'' is an increasing filtration defined by
:<math>W_n=\oplus_{p+q\le n}^{}H^{p,q}</math>.
The weight filtration is defined over the reals, while the Hodge filtration is defined over the complex numbers.  A Hodge structure is determined by its weight filtration and  Hodge filtration. -->
 
===''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 '''R''' of [[real number]]s, for which ''A''&nbsp;⊗<sub>'''Z'''</sub> '''R''' is a field. Then a pure Hodge '''A'''-structure of weight ''n'' is defined as before, replacing '''Z''' 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''<sub>''X''</sub>(''t''), called its '''virtual Poincaré polynomial''', with the properties
* If ''X'' is nonsingular and projective (or complete)
::<math>P_X(t)=\sum \text{rank}(H^n(X))t^n</math>
* If ''Y'' is closed algebraic subset of ''X'' and ''U'' = ''X''\''Y''
::<math>P_X(t)=P_Y(t)+P_U(t)</math>
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 [[Motive (algebraic geometry)|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 [[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, let us consider the case of a reducible complex algebraic curve ''X'' consisting of two nonsingular components  ''X''<sub>1</sub> and ''X''<sub>2</sub>, which transversally intersect at the points ''Q''<sub>1</sub> and ''Q''<sub>2</sub>. Further, assume that the components are not compact, but can be compactified by adding the points ''P''<sub>1</sub>, ..., ''P<sub>n</sub>''. 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 α<sub>''i''</sub> representing small loops around the punctures ''P<sub>i</sub>''. Then there are elements β<sub>''j''</sub> that are coming from the first homology of the compactification of one of the components. The lifting of a one-cycle in ''X<sub>k</sub>'', ''k''&nbsp;=&nbsp;1,&nbsp;2, to a cycle in ''X'' is not canonical: these elements are determined modulo the span of  α<sub>''i''</sub>. Finally, modulo the first two types, the group is generated by a combinatorial cycle γ which goes from ''Q''<sub>1</sub> to ''Q''<sub>2</sub> along a path in one component  ''X''<sub>1</sub> and comes back along a path in the other component ''X''<sub>2</sub>. This suggests that ''H''<sup>1</sup>(''X'') admits an increasing filtration
 
: <math> 0\subset W_0\subset W_1 \subset W_2=H^1(X),\, </math>
 
whose successive quotients ''W<sub>n</sub>''/''W''<sub>''n''−1</sub> originate from the cohomology of smooth complete varieties, hence admit (pure) Hodge structures, albeit of different weights.
 
=== Definition of mixed Hodge structure ===
A '''mixed Hodge structure''' on an abelian group ''H''<sub>'''Z'''</sub> consists of a finite decreasing filtration ''F''<sup>''p''</sup> on the complex vector space ''H'' (the complexification of ''H''<sub>'''Z'''</sub>), called the '''Hodge filtration''' and a finite increasing filtration ''W''<sub>''i''</sub> on the rational vector space ''H''<sub>'''Q'''</sub>&nbsp;=&nbsp;''H''<sub>'''Z'''</sub>&nbsp;⊗<sub>'''Z'''</sub> '''Q''' (obtained by extending the scalars to rational numbers), called the '''weight filtration''', subject to the requirement that the ''n''th associated graded quotient of ''H''<sub>'''Q'''</sub> 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
 
: <math>\operatorname{gr}_n^{W} H = W_n\otimes\mathbf{C}/W_{n-1}\otimes\mathbf{C} </math>
 
is defined by
 
: <math> F^p \operatorname{gr}_n^{W} H = (F^p\cap W_n\otimes\mathbf{C}+W_{n-1}\otimes\mathbf{C})/W_{n-1}\otimes\mathbf{C}.</math>
 
Retrospectively, one sees that the total cohomology of a compact Kähler manifold has a mixed Hodge structure, where the ''n''th space of the weight filtration ''W''<sub>''n''</sub> 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''<sup>''p''</sup> and a decreasing filtration ''W''<sub>''n''</sub> 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 '''C'''*. 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 category|Tannakian formalism]].
 
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]]. The kernels and cokernels in this category coincide with the usual kernels and cokernels in the category of vector spaces, with the induced filtrations.''
 
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 duality|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. {{Harvtxt|Deligne|1982}} <ref> The second article titled ''Tannakian categories'' by Deligne and Milne concentrated to this topic.</ref>
 
=== 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 theorem|''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 [[Hyperhomology|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.
 
==Examples==
*The '''Tate Hodge structure''' '''Z'''(1) is the Hodge structure with underlying '''Z''' module given by 2π''i'''''Z''' (a subgroup of '''C'''), with '''Z'''(1) ⊗ '''C''' = ''H''<sup>−1,−1</sup>. 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 '''Z'''(''n''); 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 {{Harvtxt|Deligne|1971}},{{Harvtxt|Deligne|1971a}} and in general by {{Harvtxt|Deligne|1974}}.
 
== Applications ==
The machinery based on the notions of Hodge structure and mixed Hodge structure forms a part of still largely conjectural theory of [[motive (algebraic geometry)|motives]] envisaged by [[Alexander Grothendieck]]. Arithmetic information for nonsingular algebraic variety ''X'', encoded by eigenvalue of [[Frobenius element]]s 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 <math>R_{\mathbf {C/R}}{\mathbf C}^*</math> on the de Rham cohomology. Since then, the mystery has deepened with the discovery and mathematical formulation of [[mirror symmetry (string theory)|mirror symmetry.]]
 
==Variation of Hodge structure==
A '''variation of Hodge structure''' ({{Harvtxt|Griffiths|1968}},{{Harvtxt|Griffiths|1968a}},{{Harvtxt|Griffiths|1970}}) 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''<sub>''X''</sub>, subject to the following two conditions:
*The filtration induces a Hodge structure of weight ''n'' on each stalk of the sheaf ''S''
*('''[[Griffiths transversality]]''') The natural connection on ''S'' ⊗ ''O''<sub>''X''</sub>  maps ''F''<sup>''n''</sup> into ''F''<sup>''n''−1</sup> ⊗ Ω<sup>1</sup><sub>''X''</sub>.
 
Here the natural (flat) connection on ''S'' ⊗ ''O''<sub>''X''</sub> induced by the flat connection on ''S'' and the flat connection ''d'' on ''O''<sub>''X''</sub>, and  ''O<sub>X</sub>'' is the sheaf of holomorphic functions on ''X'', and Ω<sup>1</sup><sub>''X''</sub> 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''.
 
==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 {{Harvtxt|Saito|1989}} 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''<sub>∗</sub>, ''f*'', ''f''<sub>!</sub>, ''f''<sup>!</sup> between ([[derived categories]] of) mixed Hodge modules similar to the ones for sheaves.
 
== See also ==
* [[Hodge conjecture]]
*[[Hodge–Tate structure]], a ''p''-adic analogue of Hodge structures.
* [[Logarithmic form]]
 
== Notes ==
<references/>
 
==References==
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Travaux de Griffiths| publisher=Sem. Bourbaki Exp. 376, Lect. notes in math. Vol 180 |  year=1971b | pages=213–235}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Actes du Congrès International des Mathématiciens (Nice, 1970) | url=http://math.harvard.edu/~tdp/Deligne-Theorie.de.Hodge-1-single-page.pdf | publisher=Gauthier-Villars | mr=0441965 | year=1971 | volume=1 | chapter=Théorie de Hodge. I | pages=425–430}} This constructs a mixed Hodge structure on the cohomology of a complex variety.
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Théorie de Hodge. II. | url=http://www.numdam.org/item?id=PMIHES_1971__40__5_0 | publisher=Inst. Hautes Études Sci. Publ. Math. No. 40 | year=1971a | mr=0498551 | pages=5–57}} This constructs a mixed Hodge structure on the cohomology of a complex variety.
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Théorie de Hodge. III. | url=http://www.numdam.org/item?id=PMIHES_1974__44__5_0 | publisher=Inst. Hautes Études Sci. Publ. Math. No. 44 | year=1974 | mr=0498552 | pages=5–77}} This constructs a mixed Hodge structure on the cohomology of a complex variety.
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Structures de Hodge mixtes réelles.  Motives (Seattle, WA, 1991) | url=http://www.numdam.org/item?id=PMIHES_1974__44__5_0 | publisher=Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. | year=1994 | mr=1265541 | pages=509–514}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties by Pierre Deligne, James S. Milne, Arthur Ogus, Kuang-yen Shih | publisher=Springer-Verlag, Lecture Notes in Math. 900 | year=1982 | pages=1–414}}.  An annotated version of this article can be found on J. Milne's [http://www.jmilne.org/math/xnotes/tc.html homepage].
*{{Citation | last1=Griffiths | first1=P. | author1-link=P. Griffiths | title=Periods of integrals on algebraic manifolds I (Construction and Properties of the Modular Varieties) | url=http://links.jstor.org/sici?sici=0002-9327%28196804%2990%3A2%3C568%3APOIOAM%3E2.0.CO%3B2-V | publisher=Amer. J. Math., 90 | year=1968 | pages=568–626}}
*{{Citation | last1=Griffiths | first1=P. | author1-link=P. Griffiths | title=Periods of integrals on algebraic manifolds II (Local Study of the Period Mapping) | url=http://links.jstor.org/sici?sici=0002-9327%28196807%2990%3A3%3C805%3APOIOAM%3E2.0.CO%3B2-2 | publisher=Amer. J. Math., 90 | year=1968a | pages=808–865}}
*{{Citation | last1=Griffiths | first1=P. | author1-link=P. Griffiths | title=Periods of integrals on algebraic manifolds III. Some global differential-geometric properties of the period mapping. | url=http://www.numdam.org/item?id=PMIHES_1970__38__125_0 | publisher=Publ. Math. IHES, 38 | year=1970 | pages=228–296}}
*{{springer|id=H/h047470|title=Hodge structure|author=A.I. Ovseevich}}
*{{Citation | last1=Saito | first1=Morihiko | author1-link=Morihiko Saito | title=Introduction to mixed Hodge modules. Actes du Colloque de Théorie de Hodge (Luminy, 1987). | publisher=Astérisque No. 179-180 | year=1989 | mr=1042805 | pages=145–162}}
*{{springer|id=V/v096170|title=Variation of Hodge structure|author=J. Steenbrink}}
 
[[Category:Homological algebra]]
[[Category:Hodge theory]]
[[Category:Structures on manifolds]]

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