---
tag: ZSM083M97C
kind: concept
name: Hodge structure
msc: [14C30, 14D07]
sources:
  - url: https://en.wikipedia.org/wiki/Hodge_structure
same_as:
  wikidata: Q4444341
  wikipedia: Hodge structure
---

## Summary

A Hodge structure of weight $n$ is a lattice, or a rational vector space, with extra data on its complexification.
The complexification splits into pieces $H^{p,q}$ with $p + q = n$, and complex conjugation exchanges $H^{p,q}$ and $H^{q,p}$.
It abstracts the structure that the [[Hodge decomposition|HABCJ4Y0SR]] puts on the cohomology of a compact Kähler manifold.
Hodge structures form an abelian category with tensor products.
They are the linear algebra data behind period maps, variations of Hodge structure and the [[Hodge conjecture|XAEEDWV3PM]].

## Explanations

### L1 human

A Hodge structure is a lattice $H_\Z$ together with a way to split the complex vector space $H_\Z \otimes \C$ into pieces, labelled by pairs of integers.
The pieces must be arranged symmetrically under complex conjugation.
The first example comes from an elliptic curve $E = \C / (\Z + \tau \Z)$, with $\Im \tau > 0$.
Its first cohomology $H^1(E, \Z)$ is a rank $2$ lattice, and over $\C$ it splits into the line of $dz$ and the line of $d\bar z$, a Hodge structure of weight $1$.
The position of the line of $dz$ relative to the lattice recovers $\tau$, and so the curve.