---
tag: STYAB4PFDW
kind: concept
name: Calabi–Yau manifold
msc: [14J32, 32Q25]
sources:
  - url: https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold
same_as:
  wikidata: Q549959
  wikipedia: Calabi–Yau manifold
---

## Summary

A Calabi–Yau manifold is a compact [[Kähler manifold|X60JG9AEQJ]] with trivial canonical bundle.
By Yau's proof of the [[Calabi conjecture|WJKA5YBTJ7]], each Kähler class on it contains a Ricci-flat Kähler metric.
Calabi–Yau manifolds sit between Fano varieties and varieties of general type in the classification of varieties.
They are central in string theory and in [[mirror symmetry|YK3DMN2NSK]].

## Explanations

### L1 human

The canonical bundle measures how a complex manifold curves on average, and on a Calabi–Yau manifold it is trivial, so the manifold is flat on average.
In complex dimension $1$ the Calabi–Yau manifolds are the elliptic curves, which are tori.
In dimension $2$ the simply connected ones are the K3 surfaces.
A smooth quintic hypersurface in $\mathbb{P}^4$ is a Calabi–Yau threefold, and it is the most studied example in physics.