complex geometry

# Contents

## Definition

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

## 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

Revised on April 11, 2014 02:06:05 by Urs Schreiber (185.37.147.12)