---
tag: Y92V5H3KJ2
kind: definition
name: Hodge structure
msc: [14C30, 14D07]
defines: ZSM083M97C
uses: [082GHNPJ53, 000AYPA9AJ, 0XYYT5KNTX, HNK1Y62G7V]
sources:
  - url: https://en.wikipedia.org/wiki/Hodge_structure
---

## Definition

Let $n$ be an integer.
A *pure Hodge structure of weight $n$* consists of:

1. a finitely generated [[abelian group|082GHNPJ53]] $H_\Z$, with complex [[vector space|000AYPA9AJ]] $H_\C = H_\Z \otimes_\Z \C$;
2. a [[direct sum|0XYYT5KNTX]] decomposition $H_\C = \bigoplus_{p + q = n} H^{p,q}$ into complex subspaces;
3. the symmetry $\overline{H^{p,q}} = H^{q,p}$ for all $p, q$, where the bar is complex conjugation on $H_\C$ with respect to the real form $H_\Z \otimes \R$.

A *rational Hodge structure* is defined in the same way with a finite-dimensional $\Q$-vector space $H_\Q$ in place of $H_\Z$.

## Remarks

Equivalently, the data can be given by the Hodge [[filtration|HNK1Y62G7V]] $F^p = \bigoplus_{p' \ge p} H^{p', n - p'}$, a decreasing filtration of $H_\C$ with $H_\C = F^p \oplus \overline{F^{n-p+1}}$ for all $p$; then $H^{p,q} = F^p \cap \overline{F^q}$.

Example: let $X$ be a compact Kähler manifold, for instance a smooth projective complex variety.
Then $H^k(X, \Z)$ modulo torsion, with the Hodge decomposition $H^k(X, \C) = \bigoplus H^{p,q}(X)$, is a Hodge structure of weight $k$: clauses 2 and 3 are parts 1 and 2 of the Hodge decomposition theorem.
The Tate structure $\Z(1)$ is $H_\Z = 2\pi i \Z \subseteq \C$ with $H_\C = H^{-1,-1}$, a Hodge structure of weight $-2$.

Non-example: let $H_\Z = \Z^2$ and $n = 1$, with $H^{1,0} = H_\C$ and $H^{0,1} = 0$.
Clause 2 holds, but clause 3 fails, since $\overline{H^{1,0}} = H_\C \ne 0 = H^{0,1}$.