nLab
K3 surface

Definition

A K3 surface is a Calabi-Yau variety of dimension $2$. This means that the canonical bundle ${\omega }_X={\wedge }^2{\Omega }_X\simeq {𝒪}_X$ is trivial and $H^1(X,{𝒪}_X)=0$.

Examples

• A cyclic cover ${\mathbb{P}}^2$ branched over a curve of degree $6$

• A nonsingular degree $4$ hypersurface in ${\mathbb{P}}^3$.

Basic Properties

• All K3 surfaces are simply connected.

• The Hodge diamond? is completely determined (even in positive characteristic) and hence the Hodge-de Rham spectral sequence? degenerates at $E_1$. This also implies that the Betti numbers are completely determined as $1,0,22,0,1$.

• Over the complex numbers they are all Kähler.

References

• Claire Voisin, Geometry of the Moduli Space of K3 surfaces

• David Morrison, The geometry of K3 surfaces Lecture notes (1988)