# Hodge structure

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 $\overline{F ^ { s } } \cap F ^ { r } = 0$ 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 $X$, which was first studied by W.V.D. Hodge (see [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), or as the cohomology space $H ^ {q} ( X, \Omega ^ {p} )$ of sheaves $\Omega ^ {p}$ of holomorphic differential forms [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 $\overline{F ^ { p } }$ determine a pure Hodge structure of weight $n$. The mixed Hodge structure in the cohomology spaces of a complex algebraic variety (not necessarily compact or smooth) is an analogue of the structure of the Galois module in the étale cohomology (cf. [3]). The Hodge structure has important applications in algebraic geometry (see Period mapping) and in the theory of singularities of smooth mappings (see [4]).

#### References

 [1] W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952) MR0051571 [2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001 [3] 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 MR0432648 Zbl 0334.14011 [4] 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 Zbl 0554.58002

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 $F ^ { p } \oplus \overline{F ^ { q } } = H _ {\mathbf C}$ 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 $0 \neq x \in F ^ { p } \cap \overline{F ^ { n- p } }$. The Hodge structures arising in algebraic geometry are always polarizable.
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