# 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 ${F ^ { s } } bar \cap F ^ { r } = 0$ 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 $X$, which was first studied by W.V.D. Hodge (see ). 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 . The Hodge filtration in $H ^ {n} ( X, \mathbf C )$ arises from the filtration of the sheaf complex $\Omega ^ {bold \cdot } = \sum _ {p \geq 0 } \Omega ^ {p}$, 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 (not necessarily compact or smooth) is an analogue of the structure of the Galois module in the étale cohomology (cf. ). The Hodge structure has important applications in algebraic geometry (see Period mapping) and in the theory of singularities of smooth mappings (see ).