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''of weight  $  n $ (pure)''

An object consisting of a lattice  $  H _ {\mathbf Z }  $
in the real vector space  $  H _ {\mathbf R }  = H _ {\mathbf Z }  \otimes \mathbf R $
and a decomposition  $  H _ {\mathbf C }  = \oplus _ {p + q = n }  H  ^ {p,q} $
of the complex vector space  $  H _ {\mathbf C }  = H _ {\mathbf Z }  \otimes \mathbf C $(
a Hodge decomposition). Here the condition  $  \overline{ {H  ^ {p,q} }}\; = H  ^ {q,p} $
must hold, where the bar denotes complex conjugation in  $  H _ {\mathbf C }  = H _ {\mathbf R }  \otimes _ {\mathbf R }  \mathbf C $. 
Another description of the Hodge decomposition consists in the specification of a decreasing filtration (a Hodge filtration)  $  F ^ { r } = \oplus _ {p \geq  r }  H  ^ {p,q} $
in  $  H _ {\mathbf C }  $
such that  $  {F ^ { s } } bar \cap F ^ { r } = 0 $
for  $  r + s \neq n $. 
Then the subspace  $  H  ^ {p,q} $
can be recovered by the formula  $  H  ^ {p,q} = F  ^ {p} \cap \overline{ {F  ^ {q} }}\; $.

An example is the Hodge structure in the  $  n $-
dimensional cohomology space  $  H  ^ {n} ( X, \mathbf C ) $
of a compact [[Kähler manifold|Kähler manifold]]  $  X $, 
which was first studied by W.V.D. Hodge (see [[#References|[1]]]). In this case the subspace  $  H  ^ {p,q} $
can be described as the space of harmonic forms of type  $  ( p, q) $(
cf. [[Harmonic form|Harmonic form]]), or as the cohomology space  $  H  ^ {q} ( X, \Omega  ^ {p} ) $
of sheaves  $  \Omega  ^ {p} $
of holomorphic differential forms [[#References|[2]]]. The Hodge filtration in  $  H  ^ {n} ( X, \mathbf C ) $
arises from the filtration of the sheaf complex  $  \Omega ^ {\bullet } = \sum _ {p \geq  0 }  \Omega  ^ {p} $, 
the  $  n $-dimensional hypercohomology group of which is  $  H  ^ {n} ( X, \mathbf C ) $, 
by subcomplexes  $  \sum _ {p \geq  r }  \Omega  ^ {r} $.

A more general concept is that of a mixed Hodge structure. This is an object consisting of a lattice  $  H _ {\mathbf Z }  $
in  $  H _ {\mathbf R }  = H _ {\mathbf Z }  \otimes \mathbf R $, 
an increasing filtration (a filtration of weights)  $  W _ {n} $
in  $  H _ {\mathbf Q }  = H _ {\mathbf Z }  \otimes \mathbf Q $
and a decreasing filtration (a Hodge filtration)  $  F  ^ {p} $
in  $  H _ {\mathbf C }  = H _ {\mathbf Z }  \otimes \mathbf C $, 
such that on the space  $  ( W _ {n+1} /W _ {n} ) \otimes \mathbf C $, 
the filtrations  $  F ^ { p } $
and  $  {F ^ { p } } bar $
determine a pure Hodge structure of weight  $  n $. 
The mixed Hodge structure in the cohomology spaces of a complex [[Algebraic variety|algebraic variety]] (not necessarily compact or smooth) is an analogue of the structure of the Galois module in the étale cohomology (cf. [[#References|[3]]]). The Hodge structure has important applications in algebraic geometry (see [[Period mapping|Period mapping]]) and in the theory of singularities of smooth mappings (see [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952) {{MR|0051571}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Deligne, "Poids dans la cohomologie des variétés algébriques" R. James (ed.) , ''Proc. Internat. Congress Mathematicians (Vancouver, 1974)'' , '''1''' , Canad. Math. Congress (1975) pp. 79–85 {{MR|0432648}} {{ZBL|0334.14011}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.N. Varchenko, "Asymptotic integrals and Hodge structures" ''J. Soviet Math.'' , '''27''' (1984) pp. 2760–2784 ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''22''' (1983) pp. 130–166 {{MR|}} {{ZBL|0554.58002}} </TD></TR></table>

See also the references to [[Period mapping|Period mapping]].

====Comments====
A Hodge structure of weight  $  n $
thus consists of i) a finitely-generated Abelian group  $  H _ {\mathbf Z} $; 
and ii) a finite decreasing filtration  $  F ^ { \bullet } $
on  $  H _ {\mathbf C} = H _ {\mathbf Z} \otimes \mathbf C $
such that  $  F ^ { p } \oplus {F ^ { q } } bar = H _ {\mathbf C} $
as soon as  $  p + q = n + 1 $. 
A polarization of a Hodge structure of weight  $  n $
is a  $  ( - 1 )  ^ {n} $-
symmetric  $  \mathbf Z $-
valued bilinear form  $  S $
on  $  H _ {\mathbf Z} $
such that  $  S ( x , y) = 0 $
for  $  x \in F ^ { p } $, 
$  y \in F ^ { n- p- 1 } $
and such that  $  i  ^ {p-q} S ( x , \overline{x}\; ) > 0 $
for  $  0 \neq x \in F ^ { p } \cap {F ^ { n- p } } bar $. 
The Hodge structures arising in algebraic geometry are always polarizable.

There exist classifying spaces for polarized Hodge structures with given numerical data [[#References|[a1]]], as well as for mixed Hodge structures with a polarization on graded quotients for the weight filtration [[#References|[a2]]]. Mixed Hodge structures form an [[Abelian category|Abelian category]] in which every morphism is strictly compatible with both Hodge and weight filtrations. Pure polarized Hodge structures form a Tannakian category [[#References|[a3]]]. There exist canonical and functorial mixed Hodge structures on (local) cohomology groups of algebraic varieties , rational homotopy groups [[#References|[a5]]], vanishing cycle groups of function germs [[#References|[a6]]], [[#References|[a7]]], and on the intersection homology groups of algebraic varieties with coefficients in a polarizable variation of Hodge structure [[#References|[a8]]], [[#References|[a9]]]. In the latter case, there is even a pure Hodge structure. At this moment (1989), the ultimate generalization seems to be the concept of a mixed Hodge module [[#References|[a10]]]–[[#References|[a11]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Griffiths, "Periods of integrals on algebraic manifolds" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–625; 805–865 {{MR|0242844}} {{MR|0233825}} {{MR|0229641}} {{ZBL|0188.24801}} {{ZBL|0183.25501}} {{ZBL|0169.52303}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Carlson, E. Cattani, A. Kaplan, "Mixed Hodge structures and compactifications of Siegel's space" A. Beauville (ed.) , ''Algebraic geometry (Angers, 1979)'' , Sijthoff &amp; Noordhoff (1980) pp. 77–105 {{MR|605337}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> N. Saavedra Rivano, "Catégories Tannakiennes" , ''Lect. notes in math.'' , '''265''' , Springer (1972) {{MR|}} {{ZBL|0246.14003}} {{ZBL|0241.14008}} </TD></TR><TR><TD valign="top">[a4a]</TD> <TD valign="top"> P. Deligne, "Théorie de Hodge II, III" ''Publ. Math. IHES'' , '''40''' (1971) pp. 5–58 {{MR|498551}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4b]</TD> <TD valign="top"> P. Deligne, "Théorie de Hodge IV" ''Publ. Math. IHES'' , '''44''' (1975) pp. 5–77</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.M. Hain, "The de Rham homotopy theory of complex algebraic varieties I, II" ''$K$-theory'' , '''1''' (1987) pp. 271–324; 481–497</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.H.M. Steenbrink, "Mixed Hodge structure on the vanishing cohomology" P. Holm (ed.) , ''Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School'' , Sijthoff &amp; Noordhoff (1977) pp. 525–563 {{MR|0485870}} {{ZBL|0373.14007}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> V. Navarro Aznar, "Sur la théorie de Hodge–Deligne" ''Invent. Math.'' , '''90''' (1987) pp. 11–76 {{MR|}} {{ZBL|0639.14002}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> E. Cattani, A. Kaplan, W. Schmid, "$L^2$ and intersection cohomologies for a polarizable variation of Hodge structure" ''Invent. Math.'' , '''87''' (1987) pp. 217–252 {{MR|870728}} {{ZBL|0611.14006}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Kashiwara, T. Kawai, "The Poincaré lemma for variations of Hodge structure" ''Publ. RIMS Kyoto Univ.'' , '''23''' : 2 (1987) pp. 345–407 {{MR|}} {{ZBL|0629.14005}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Saito, "Modules de Hodge polarisables" ''Preprint RIMS'' , '''553''' (Oct. 1986) {{MR|1000123}} {{ZBL|0691.14007}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> M. Saito, "Mixed Hodge modules" ''Preprint RIMS'' , '''585''' (July 1987) {{MR|2843222}} {{MR|1308540}} {{MR|1159259}} {{MR|1047741}} {{MR|1047415}} {{MR|1054250}} {{MR|1042805}} {{MR|0888149}} {{MR|0888148}} {{ZBL|05986955}} {{ZBL|0826.32029}} {{ZBL|0815.14008}} {{ZBL|0727.14004}} {{ZBL|0726.14007}} {{ZBL|0753.32004}} {{ZBL|0765.14006}} {{ZBL|0635.14009}} {{ZBL|0635.14008}} </TD></TR></table>