---
tag: 32YJXG30MF
kind: concept
name: Hodge theory
msc: [14C30, 58A14, 32J25]
sources:
  - url: https://en.wikipedia.org/wiki/Hodge_theory
same_as:
  wikidata: Q1317202
  wikipedia: Hodge theory
---

## Summary

Hodge theory studies the cohomology of smooth manifolds through harmonic differential forms.
For complex algebraic varieties, it studies the extra structure that harmonic forms put on cohomology.
On a compact Riemannian manifold, the [[Hodge theorem|A127WFVYKW]] identifies [[de Rham cohomology|GX74C07T0G]] with the space of harmonic forms.
On a compact [[Kähler manifold|X60JG9AEQJ]], harmonic forms split by type, which gives the [[Hodge decomposition|HABCJ4Y0SR]] and a [[Hodge structure|ZSM083M97C]] on each cohomology group.
The theory gives strong topological restrictions on algebraic varieties and leads to deep open problems such as the [[Hodge conjecture|XAEEDWV3PM]].

## Explanations

### L1 human

A cohomology class on a manifold can be represented by many differential forms.
Hodge theory picks the best one, the harmonic form, which solves a Laplace equation and minimizes energy.
On complex manifolds with a compatible metric, harmonic forms can be sorted by how many $dz$ and $d\bar{z}$ factors they contain.
For a compact Riemann surface of genus $g$, this sorts the $2g$-dimensional first cohomology into $g$ holomorphic forms and $g$ antiholomorphic ones.
Such facts constrain the shape of algebraic varieties, for instance the odd Betti numbers of a smooth projective variety are even.