Hodge structure

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In mathematics, a Hodge structure is a structure similar to the one that Hodge theory gives to the cohomology of a smooth compact Kähler manifold. A mixed Hodge structure is a generalization found by Deligne (1970) that extands this to the case of all complex varieties (possibly singular and non-complete).

1 Motivation
2 Definitions
3 The mixed Hodge structure of a complex algebraic variety
4 External links
5 References

[edit] Motivation

[edit] Definitions

A real Hodge structure consists of a finite dimensional real vector space H with a decomposition of H⊗C as a direct sum of complex subspaces

$H\otimes {\mathbb C} = \oplus_{p,q\in Z}H^{p,q},$

such that H^p,q is the complex conjugate of H^q,p. If H^p,q = 0 for p+q≠ n, the real Hodge structure is said to be pure (of weight n). For example, by Hodge theory the real cohomology of a compact Kähler manifold is a real Hodge structure, and the nth real cohomology group is pure of weight n.

A real Hodge structure can also be defined as a representation of the 2-dimensional commutative real algebraic group G defined at the base change of the multiplicative group from R to C; in other words, if A is an algebra over R, then the group G(A) of A-valued points of G is the multiplicative group of A⊗C. In particular G(R) is the group C^* of non-zero complex numbers. The subspace H^p,q is the subspace on which z∈C^* acts as multiplication by $z^p\overline{z}^q$

A Hodge structure is graded as the sum of Hodge structures of weight n for integers n. Given a Hodge structure, the Hodge filtration F is a decreasing filtration defined by

$F^p=\oplus_{p'\ge p, q'\in Z}^{}H^{p',q'}$

and the weight filtration W is an increasing filtration defined by

$W_n=\oplus_{p+q\le n}^{}H^{p,q}$ .

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 mixed real Hodge structure consists of a finite dimensional real vector space H with an increasing filtration W on H (called the weight filtration) and a decreasing filtration F on H⊗C (called the Hodge filtration) such that the Hodge filtration F induces a Hodge structure on the graded space

$Gr_W = \oplus_{n\in Z}W_n/W_{n-1}$

of the filtration, such that the grading is the weight grading of the Hodge structure.

A mixed Hodge structure over the integers consists of a finitely generated module H over the integers with an increasing filtration W on H⊗Q and a decreasing filtration F on H⊗C that induce a mixed real Hodge structure on the real vector space H⊗R. A rational Hodge structure is defined in the same way, with the integers replaced by the rational numbers.

Mixed Hodge structures form an abelian category.

[edit] The mixed Hodge structure of a complex algebraic variety

[edit] External links

Mixed hodge structures by L. I. Nicolaescu

[edit] References

Deligne, Pierre Théorie de Hodge. I. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 425-430. Gauthier-Villars, Paris, 1971. MR 0441965 This defines mixed Hodge structures.
Deligne, Pierre Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. No. 40 (1971), 5-57.MR 0498551 This constructs a mixed Hodge structure on the cohomology of a non-singular complex variety.
Deligne, Pierre Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. No. 44 (1974), 5--77. MR 0498552 This constructs a mixed Hodge structure on the cohomology of a complex variety.
Deligne, Pierre Structures de Hodge mixtes réelles. Motives (Seattle, WA, 1991), 509-514, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. MR 1265541
A.I. Ovseevich, "Hodge structure" SpringerLink Encyclopaedia of Mathematics (2001)
Saito, Morihiko Introduction to mixed Hodge modules. Actes du Colloque de Théorie de Hodge (Luminy, 1987). Astérisque No. 179-180 (1989), 10, 145-162. MR 1042805