Hodge structure

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 groups 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 singular and non-complete). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989). 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 groups 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 singular and non-complete). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989). A pure Hodge structure of integer weight n consists of an abelian group H Z {displaystyle H_{mathbb {Z} }} and a decomposition of its complexification H into a direct sum of complex subspaces Hp,q, where p + q = n, with the property that the complex conjugate of Hp,q is Hq,p: 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 F p H ( p ∈ Z ) , {displaystyle F^{p}H(pin mathbb {Z} ),} subject to the condition