---
tag: Q3V566R6K2
kind: concept
name: K3 surface
msc: [14J28]
specializes: [KFTECAATKG, STYAB4PFDW]
sources:
  - url: https://en.wikipedia.org/wiki/K3_surface
same_as:
  wikidata: Q1969721
  wikipedia: K3 surface
---

## Summary

A K3 surface is a smooth projective [[algebraic surface|KFTECAATKG]], or more generally a compact complex surface, with trivial canonical bundle and no holomorphic $1$-forms.
K3 surfaces are the two-dimensional [[Calabi–Yau manifolds|STYAB4PFDW]], the surface analogue of elliptic curves.
All complex K3 surfaces are diffeomorphic to each other and simply connected, with second Betti number $22$.
They are central in the classification of surfaces, in arithmetic geometry and in string theory.

## Explanations

### L1 human

The simplest K3 surface is a smooth surface of degree $4$ in projective three-space, such as $x^4 + y^4 + z^4 + w^4 = 0$.
Its canonical bundle is trivial, which means that it carries a nowhere vanishing holomorphic $2$-form, a kind of volume form.
This makes K3 surfaces flat in an averaged sense, between the positively curved rational surfaces and the negatively curved surfaces of general type.