complex geometry

Contents

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 ${H}^{1}\left(X,{𝒪}_{X}\right)=0$.

Examples

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

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

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

Revised on November 6, 2012 18:53:14 by Urs Schreiber (131.174.188.92)