Pure Hodge structure - Imported Knowlpedia Content

Definition

A pure real Hodge structure of weight $n$ is a finite-dimensional real vector space $V_{\mathbb R}$ together with a direct-sum decomposition of its complexification

V_{\mathbb C}=V_{\mathbb R}\otimes_{\mathbb R}\mathbb C =\bigoplus_{p+q=n}V^{p,q}

such that complex conjugation relative to $V_{\mathbb R}$ satisfies $\overline{V^{p,q}}=V^{q,p}$ . The integers $p$ and $q$ may be negative unless an effectiveness condition is imposed. A pure rational Hodge structure is defined in the same way from a finite-dimensional $\mathbb Q$ -vector space, and an integral Hodge structure starts with a finite-rank free abelian group.

Filtration formulation

The decomposition determines the decreasing Hodge filtration

F^pV_{\mathbb C}=\bigoplus_{r\geq p}V^{r,n-r}.

Conversely, a filtration defines a pure real Hodge structure of weight $n$ exactly when

V_{\mathbb C}=F^pV_{\mathbb C}\oplus\overline{F^{\,n-p+1}V_{\mathbb C}}

for every $p$ . Then $V^{p,q}=F^p\cap\overline{F^q}$ Voisin, §7.1.1.

Geometric example

For a compact Kähler manifold $X$ , the decomposition

H^k(X,\mathbb C)=\bigoplus_{p+q=k}H^{p,q}(X)

makes $H^k(X,\mathbb R)$ a pure real Hodge structure of weight $k$ . The integral lattice $H^k(X,\mathbb Z)$ modulo torsion gives an integral Hodge structure. The conjugation condition follows from conjugation of differential forms.

Conventions and scope

“Pure” means that every bidegree has the same sum $p+q=n$ ; it does not mean that only one bidegree occurs. A mixed Hodge structure has an additional increasing weight filtration whose graded pieces are pure. Authors sometimes build a chosen lattice into the phrase “Hodge structure”; the coefficient ring should therefore always be stated.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Chapter record. Relevant: §7.1.1, Hodge structures and their filtration description.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: the appendix on Hodge theory and Chapter 3 on compact Kähler cohomology.